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Trigonometry Problems - sin, cos, tan, cot: Problems with Solutions

Trigonometry Questions

Trigonometry questions given here involve finding the missing sides of a triangle with the help of trigonometric ratios and proving trigonometry identities. We know that trigonometry is one of the most important chapters of Class 10 Maths. Hence, solving these questions will help you to improve your problem-solving skills.

What is Trigonometry?

The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle.

The basic trigonometric ratios are defined as follows.

sine of ∠A = sin A = Side opposite to ∠A/ Hypotenuse

cosine of ∠A = cos A = Side adjacent to ∠A/ Hypotenuse

tangent of ∠A = tan A = (Side opposite to ∠A)/ (Side adjacent to ∠A)

cosecant of ∠A = cosec A = 1/sin A = Hypotenuse/ Side opposite to ∠A

secant of ∠A = sec A = 1/cos A = Hypotenuse/ Side adjacent to ∠A

cotangent of ∠A = cot A = 1/tan A = (Side adjacent to ∠A)/ (Side opposite to ∠A)

Also, tan A = sin A/cos A

cot A = cos A/sin A

Also, read: Trigonometry

Trigonometry Questions and Answers

1. From the given figure, find tan P – cot R.

From the given,

In the right triangle PQR, Q is right angle.

By Pythagoras theorem,

PR 2 = PQ 2 + QR 2

QR 2 = (13) 2 – (12) 2

= 169 – 144

tan P = QR/PQ = 5/12

cot R = QR/PQ = 5/12

So, tan P – cot R = (5/12) – (5/12) = 0

2. Prove that (sin 4 θ – cos 4 θ +1) cosec 2 θ = 2

L.H.S. = (sin 4 θ – cos 4 θ +1) cosec 2 θ

= [(sin 2 θ – cos 2 θ) (sin 2 θ + cos 2 θ) + 1] cosec 2 θ

Using the identity sin 2 A + cos 2 A = 1,

= (sin 2 θ – cos 2 θ + 1) cosec 2 θ

= [sin 2 θ – (1 – sin 2 θ) + 1] cosec 2 θ

= 2 sin 2 θ cosec 2 θ

= 2 sin 2 θ (1/sin 2 θ)

3. Prove that (√3 + 1) (3 – cot 30°) = tan 3 60° – 2 sin 60°.

LHS = (√3 + 1)(3 – cot 30°)

= (√3 + 1)(3 – √3)

= 3√3 – √3.√3 + 3 – √3

= 2√3 – 3 + 3

RHS = tan 3 60° – 2 sin 60°

= (√3) 3 – 2(√3/2)

= 3√3 – √3

Therefore, (√3 + 1) (3 – cot 30°) = tan 3 60° – 2 sin 60°.

Hence proved.

4. If tan(A + B) = √3 and tan(A – B) = 1/√3 ; 0° < A + B ≤ 90°; A > B, find A and B.

tan(A + B) = √3

tan(A + B) = tan 60°

A + B = 60°….(i)

tan(A – B) = 1/√3

tan(A – B) = tan 30°

A – B = 30°….(ii)

Adding (i) and (ii),

A + B + A – B = 60° + 30°

Substituting A = 45° in (i),

45° + B = 60°

B = 60° – 45° = 15°

Therefore, A = 45° and B = 15°.

5. If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

sin 3A = cos(A – 26°); 3A is an acute angle

cos(90° – 3A) = cos(A – 26°) {since cos(90° – A) = sin A}

⇒ 90° – 3A = A – 26

⇒ 3A + A = 90° + 26°

⇒ 4A = 116°

⇒ A = 116°/4

6. If A, B and C are interior angles of a triangle ABC, show that sin (B + C/2) = cos A/2.

We know that, for a given triangle, the sum of all the interior angles of a triangle is equal to 180°

A + B + C = 180° ….(1)

B + C = 180° – A

Dividing both sides of this equation by 2, we get;

⇒ (B + C)/2 = (180° – A)/2

⇒ (B + C)/2 = 90° – A/2

Take sin on both sides,

sin (B + C)/2 = sin (90° – A/2)

⇒ sin (B + C)/2 = cos A/2 {since sin(90° – x) = cos x}

7. If tan θ + sec θ = l, prove that sec θ = (l 2 + 1)/2l.

tan θ + sec θ = l….(i)

We know that,

sec 2 θ – tan 2 θ = 1

(sec θ – tan θ)(sec θ + tan θ) = 1

(sec θ – tan θ) l = 1 {from (i)}

sec θ – tan θ = 1/l….(ii)

tan θ + sec θ + sec θ – tan θ = l + (1/l)

2 sec θ = (l 2 + 1)l

sec θ = (l 2 + 1)/2l

8. Prove that (cos A – sin A + 1)/ (cos A + sin A – 1) = cosec A + cot A, using the identity cosec 2 A = 1 + cot 2 A.

LHS = (cos A – sin A + 1)/ (cos A + sin A – 1)

Dividing the numerator and denominator by sin A, we get;

= (cot A – 1 + cosec A)/(cot A + 1 – cosec A)

Using the identity cosec 2 A = 1 + cot 2 A ⇒ cosec 2 A – cot 2 A = 1,

= [cot A – (cosec 2 A – cot 2 A) + cosec A]/ (cot A + 1 – cosec A)

= [(cosec A + cot A) – (cosec A – cot A)(cosec A + cot A)] / (cot A + 1 – cosec A)

= cosec A + cot A

9. Prove that: (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A)

[Hint: Simplify LHS and RHS separately]

LHS = (cosec A – sin A)(sec A – cos A)

= (cos 2 A/sin A) (sin 2 A/cos A)

= cos A sin A….(i)

RHS = 1/(tan A + cot A)

= (sin A cos A)/ (sin 2 A + cos 2 A)

= (sin A cos A)/1

= sin A cos A….(ii)

From (i) and (ii),

i.e. (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A)

10. If a sin θ + b cos θ = c, prove that a cosθ – b sinθ = √(a 2 + b 2 – c 2 ).

a sin θ + b cos θ = c

Squaring on both sides,

(a sin θ + b cos θ) 2 = c 2

a 2 sin 2 θ + b 2 cos 2 θ + 2ab sin θ cos θ = c 2

a 2 (1 – cos 2 θ) + b 2 (1 – sin 2 θ) + 2ab sin θ cos θ = c 2

a 2 – a 2 cos 2 θ + b 2 – b 2 sin 2 θ + 2ab sin θ cos θ = c 2

a 2 + b 2 – c 2 = a 2 cos 2 θ + b 2 sin 2 θ – 2ab sin θ cos θ

a 2 + b 2 – c 2 = (a cos θ – b sin θ ) 2

⇒ a cos θ – b sin θ = √(a 2 + b 2 – c 2 )

Video Lesson on Trigonometry

Practice Questions on Trigonometry

Solve the following trigonometry problems.

• Prove that (sin α + cos α) (tan α + cot α) = sec α + cosec α.
• If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
• If sin θ + cos θ = √3, prove that tan θ + cot θ = 1.
• Evaluate: 2 tan 2 45° + cos 2 30° – sin 2 60°
• Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

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Trigonometric Word Problems

In these lessons, examples, and solutions we will learn the trigonometric functions (sine, cosine, tangent) and how to solve word problems using trigonometry.

Related Pages Trigonometry Word Problems Lessons On Trigonometry Inverse trigonometry Trigonometry Worksheets

The following diagram shows how SOHCAHTOA can help you remember how to use sine, cosine, or tangent to find missing angles or missing sides in a trigonometry problem. Scroll down the page for examples and solutions.

How To Solve Trigonometry Problems Or Questions?

Step 1: If no diagram is given, draw one yourself. Step 2: Mark the right angles in the diagram. Step 3: Show the sizes of the other angles and the lengths of any lines that are known. Step 4: Mark the angles or sides you have to calculate. Step 5: Consider whether you need to create right triangles by drawing extra lines. For example, divide an isosceles triangle into two congruent right triangles. Step 6: Decide whether you will need the Pythagorean theorem, sine, cosine or tangent. Step 7: Check that your answer is reasonable. The hypotenuse is the longest side in a right triangle.

How To Use Cosine To Calculate The Side Of A Right Triangle?

Solution: Use the Pythagorean theorem to evaluate the length of PR.

How To Use Tangent To Calculate The Side Of A Triangle?

Calculate the length of the side x, given that tan θ = 0.4

How To Use Sine To Calculate The Side Of A Triangle?

Calculate the length of the side x, given that sin θ = 0.6

How To Solve Word Problems Using Trigonometry?

The following video shows how to use the trigonometric ratio, tangent, to find the height of a balloon.

How To Solve Word Problems Using Sine?

This video shows how to use the trigonometric ratio, sine, to find the elevation gain of a hiker going up a slope.

Example: A hiker is hiking up a 12 degrees slope. If he hikes at a constant rate of 3 mph, how much altitude does he gain in 5 hours of hiking?

How To Use Cosine To Solve A Word Problem?

Example: A ramp is pulled out of the back of truck. There is a 38 degrees angle between the ramp and the pavement. If the distance from the end of the ramp to to the back of the truck is 10 feet. How long is the ramp? Step 1: Find the values of the givens. Step 2: Substitute the values into the cosine ratio. Step 3: Solve for the missing side. Step 4: Write the units

How To Solve Word Problems Using Tangent?

The following video shows how to use trigonometric ratio, tangent, to find the height of a building.

How To Solve Trigonometry Word Problems Using Tangent?

Example: Neil sees a rocket at an angle of elevation of 11 degrees. If Neil is located at 5 miles from the rocket launch pad, how high is the rocket?

How To Determining The Speed Of A Boat Using Trigonometry?

Example: A balloon is hovering 800 ft above a lake. The balloon is observed by the crew of a boat as they look upwards at an angle of 0f 20 degrees. 25 seconds later, the crew had to look at an angle of 65 degrees to see the balloon. How fast was the boat traveling?

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Trigonometry practice problems

Try solving these as much as you can on your own, and if you need help, look at the hidden solutions. You may use a calculator. You can download a copy of all these questions (Acrobat (PDF) 108kB Jul25 09) to use as you try these on your own. If you are having difficulty, try the Basic Trig Functions sample problems page.

Calculating the length of a side

Length of a path up a hill.

`text{Sin}=\frac{text{Opposite}}{text{Hypotenuse}`

Substituting in the appropriate values,

`text{Sin(12)}=\frac{500}{text{H}}`

`\frac{500}{text{sin(12)}}=text{H}`

sin(12)=0.21 so, `text{H}=\frac{500}{0.21}=2400  m`

The trail up the hill is 2400 m long .

Depth to a bed of coal

`text{Tan}=\frac{text{Opposite}}{text{Adjacent}}`

`text{tan(12)}=\frac{text{O}}{6\ km}`

Rearranging to isolate O,

O = tan(12) * 6 km

Using a calculator, the value of tan(12) is 0.213. So

O = 0.213 * 6 km

So the opposite side of the triangle, the depth of the coal bed, is 1.275 km , or 1275 meters .

Calculating radius of the outer core (seismology)  

`text{Cos}=\frac{text{Adjacent}}{text{Hypotenuse}}`

Substituting in,

`text{cos(52.5)}=\frac{text{A}}{6370\ km}`

Rearranging for A,

A = cos(52.5) * 6370 km so A = 3877 km

So the approximate radius of the core is 3877 km .  Note that this is approximate, because of the bending of the seismic waves as they reflect through the mantle. The actual radius is about 3500 km.

Calculating an angle

Stream gradient.

`2700\ ft\times\frac{1\ mi}{5280\ ft}=0.511\ mi`

(Note - if you need help with this step you can go to the unit conversions page ) Now we can use the formula for sine to calculate the angle x

substituting in,

`text{sin(x)}=\frac{0.511\ mi}{270\ mi}`

sin(x)= 0.00189

to solve for x, the angle, take the inverse sine of each

sin -1 (sin(x)) = sin -1 (0.00189)

since sin -1 (sin(x)) = x , our result is

x = 0.11 degrees

So the slope of the Colorado River is 0.11 degrees.

Angle of Repose

`text{tan(x)}=\frac{11\ cm}{16\ cm}`

tan(x) = .6875

tan -1 (tan(x)) = tan -1 (.6875)

since tan -1 (tan(x)) = x

x = 34.5 degrees , the angle of repose for this sand pile.

Plate tectonics and the angle of subduction

`text{tan(x)}=\frac{100\ km}{200\ km}`

Dividing, we find that

tan(x) = 0.5

tan -1 (tan(x)) = tan -1 (0.5)

x = 26.6 degrees , the angle of subduction.

TAKE THE QUIZ!!  

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Trigonometry Worksheets

Free worksheets with answer keys.

Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

(This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)

• Law of Sines
• Ambiguous Case of the Law of Sines
• Law Of Cosines
• Sine, Cosine, Tangent, to Find Side Length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
• Mixed Review
• Vector Worksheet
• Unit Circle Worksheet
• Graphing Sine and Cosine Worksheet

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Free Trigonometry Questions and Problems

Free tutorials and problems on solving trigonometric equations, trigonometric identities and formulas can also be found. Java applets are used to explore, interactively, important topics in trigonometry such as graphs of the 6 trigonometric functions, inverse trigonometric functions, unit circle, angle and sine law.

Angles in Trigonometry

• Find the coterminal angle . An analytical tutorial on how to find coterminal angles.
• Find The Reference Angle. Analytical tutorial to find the reference angle to an angle.
• Table for the 6 trigonometric functions for special angles. A table of values of sin, cos, tan, csc, sec and cot for special angles 0, 30, 45, 60 and 90 degrees.
• Special Angles On Unit Circle. Special angles together with their sine and cosine displayed on a unit circle.
• Angle in Trigonometry . Understand the definition and properties of an angle in standard position
• Special Right Triangle used to find the six trigonomatric funtions of the special angles.

Trigonometric Functions

• Periods Of Trigonometric Functions . The periods of all 6 trigonometric functions are explored interactively using an applet.
• Properties of The Six Trigonometric Functions . The properties of the 6 trigonometric functions sin (x), cos (x), tan (x), cot (x), sec (x) and csc (x) are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points.
• Sine Function . The sine function f(x) = a*sin(bx+c)+d is explored, interactively, using a large applet.
• Cosine Function . An applet helps you explore the general cosine function f(x) = a*cos(bx + c) + d.
• Tangent Function . The tangent function f(x) = a*tan(bx+c)+d and its properties such as graph, period, phase shift and asymptotes by changing the parameters a, b, c and d are explored interactively using an applet.
• Secant Function . The secant function f(x) = a*sec(bx+c)+d and its properties such as period, phase shift, asymptotes domain and range are explored using an interactive applet by changing the parameters a, b, c and d.
• Cosecant Function . The cosecant function f(x) = a * csc ( b x + c) + d and its period, phase shift, asymptotes, domain and range are explored using an applet.
• Cotangent Function . The cotangent function f(x) = a * cot ( b x + c) + d is explored along with its properties such as period, phase shift, asymptotes, domain and range.
• Graph of Sine, a*sin(bx+c), Function . Graphing and sketching sine functions of the form f (x) = a*sin (bx + c; step by step tutorial.
• Graphs of Basic Trigonometric Functions . The graphs and properties such as domain, range, vertical asymptotes of the 6 basic trigonometric functions: sin(x), cos(x), tan(x), cot(x), sec(x) and csc(x) are explored using an applet.
• Sum of Sine and Cosine Functions . An interactive tutorial to explore the sums involving sine and cosine functions such as f(x) = a*sin(bx)+ d*cos(bx).

Unit Circle in Trigonometry

• Unit Circle And The Trigonometric Functions sin(x), cos(x) and tan(x) . Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions.

Inverse Trigonometric Functions

• Inverse Trigonometric Functions . Inverse trigonometric functions are explored interactively using an applet.
• Graph, Domain and Range of Arccos function . The graph and the properties of the inverse trigonometric function arcsin are explored using an app.
• Graph, Domain and Range of Arctan function . The graph of the inverse trigonometric function arctan and its properties are explored using an app.
• Graph, Domain and Range of Arcsin function . The graph and the properties of the inverse trigonometric function arcsin are explored using an app.
• Solve Inverse Trigonometric Functions Questions . Questions on inverse trigonometric functions are solved and detailed solutions are presented. Also included are exercises with answers.
• Find Domain and Range of Arcsine Functions
• Find Domain and Range of Arccos Functions
• Trigonometry Angle Questions With Answers . Trigonometry questions related to angles in standard position, coterminal angles, complementary and supplementary angles, as well as conversion from degrees to radians and vice versa, are presented. The solutions and answers are provided.
• Rotation, Angular and Linear Speed - Questions with Answers . Questions related to angular and linear speeds of rotating objects are presented. The solutions and answers are also provided.
• Trigonometric Functions - Questions With Answers . Solve trigonometry questions related to trigonometric functions. The solutions and answers are provided.
• Simplify Trigonometric Expressions - Questions With Answers . Use trigonometric identities and formulas to simplify trigonometric expressions.
• Find Exact Values of Trigonometric Functions - Questions With Answers . Find exact values of trigonometric functions without using a calculator.

Solving Trigonometric Equations

• Trigonometric Equations . Tutorial with detailed explanations on how to solve trigonometric equations using different methods and strategies and the properties of trigonometric functions and identities.

Trigonometry Problems

• Solve Trigonometry Problems . A set of problems with detailed solutions are presented.
• Use Sine Functions to Model Problems . Tutorial on how to use sine functions to model problems. Given data and information about a certain situation, we model it in the form f(x) = A sin (b x + c) + D or f(x) = A cos (b x + c) + D .
• Solve Problems Using Trigonometric Ratios . A set of problems with detailed solutions are presented. These problems have been designed to reinforce the use of the trigonometric ratios and Pythagorean theorem.
• Tutorial on Sine Functions (1)- Problems . Tutorial on sine function problems. Examples with detailed solutions and explanations are included.
• Tutorial on Sine Functions (2)- Problems. This is a tutorial on the relationship between the amplitude, the vertical shift and the maximum and minimum of the sine function.

Trigonometric Identities and Their Applications

• Trigonometric Identities . A list of the basic trigonometric identities.
• Using Trigonometric Identities . How basic trigonometric identities are used? A tutorial with Several examples with detailed solutions is presented.
• Verify Trigonometric Identities . How to verify trigonometric identities? Several examples with detailed solutions are presented.

Trigonometric Formulas and Their Applications

• Sum, Difference and Product of Trigonometric Formulas Questions
• Sine and Cosine Sum of Angles From Euler's Formula

Polar Coordinates

• Convert Polar to Rectangular Coordinates and Vice Versa . Problems, with detailed solutions, where polar coordinates are converted into rectangular coordinates and vice versa are presented.

Problems and Self Tests

• Solve Trigonometric Equations . 10 problems, with their answers, on solving trigonometric equations are presented. These may be used as a self test on solving trigonometric equations.
• Test on Graphs Trigonometric Functions . A set of questions, with their answers, on identifying the graphs of trigonometric functions sin(x), cos(x), tan(x), sec(x), csc(x), cot(x) are presented. These may be used as a self test on the graphs of trigonometric functions.

Applications Of Trigonometry

• Sine Law - Ambiguous Case - applet . The ambiguous case of the sine law, in solving triangle problems, is explored interactively using an applet.
• Solve Right Triangle Problems with detailed solutions and explanations included.

Trigonometric Tables, Formulas and Worksheets

• Trigonometric tables . Trigonometric tables of all 6 trigonometric functions, with angles in degrees and radians. Copies of these tables can be downloaded.
• Trigonometric Identities and Formulas . Important definitions, identities and formulas used in trigonometry.
• Trigonometry Calculators . Several online trigonometry calculators and solvers in this site.
• Free trigonometry worksheets to download

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15 Trigonometry Questions And Practice Problems (KS3 & KS4): Harder GCSE Exam Style Questions Included

Beki Christian

Trigonometry questions address the relationship between the angles of a triangle and the lengths of its sides. By using our knowledge of the rules of trigonometry we can calculate missing angles or sides when we have been given some of the information. 

Here we’ve provided 15 trigonometry questions to provide students with practice at the various sorts of trigonometry problems and GCSE exam style questions you can expect in KS3 and KS4 trigonometry.

Trigonometry in the real world

Trigonometry is used by architects, engineers, astronomers, crime scene investigators, flight engineers and many others.

Trigonometry in KS3 and KS4

In KS3 we learn about the trigonometric ratios sin, cos and tan and how we can use these to calculate sides and angles in right angled triangles. In KS4 trigonometry involves applying this to a variety of situations as well as learning the exact values of sin, cos and tan for certain angles.

In the higher GCSE syllabus we learn about the sine rule, the cosine rule, a new formula for the area of a triangle and we apply trigonometry to 3D shapes. In A Level maths trigonometry is developed further but that is not the focus of the trigonometry questions here.

GCSE MATHS 2024: STAY UP TO DATE Join our email list to stay up to date with the latest news, revision lists and resources for GCSE maths 2024. We’re analysing each paper during the course of the 2024 GCSEs in order to identify the key topic areas to focus on for your revision. Thursday 16th May 2024: GCSE Maths Paper 1 2024 Analysis & Revision Topic List Monday 3rd June 2024: GCSE Maths Paper 2 2024 Analysis & Revision Topic List Monday 10th June 2024: GCSE Maths Paper 3 2024 Analysis GCSE 2024 dates GCSE 2024 results GCSE results 2023

How to answer trigonometry questions

The way to answer trigonometry questions depends on whether it is a right angled triangle or not.

Download this 15 Trigonometry Questions And Practice Problems (KS3 & KS4) Worksheet

Help your students prepare for their Maths GSCE with this free Trigonometry worksheet of 15 multiple choice questions and answers.

How to answer trigonometry questions: right-angled triangles

If your trigonometry question involves a right angled triangle, you can apply the following relationships ie SOH, CAH, TOA

To answer the trigonometry question:

1 . Establish that it is a right angled triangle.

2 . Label the opposite side (opposite the angle) the adjacent side (next to the angle) and the hypotenuse (longest side opposite the right angle).

3. Use the following triangles to help us decide which calculation to do:

How to answer trigonometry questions – non-right angled triangles

If the triangle is not a right angled triangle then we need to use the sine rule or the cosine rule.

There is also a formula we can use for the area of a triangle, which does not require us to know the base and height of the triangle.

• Establish that it is not a right angled triangle.
• Label the sides of the triangle using lower case a, b, c.
• Label the angles of the triangle using upper case A, B and C.
• Opposite sides and angles should use the same letter so for example angle C is opposite to side c.

KS3 trigonometry questions

In KS3 trigonometry questions focus on understanding of sin, cos and tan (SOHCAHTOA) to calculate missing sides and angles in right triangles.

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KS3 trigonometry questions – missing side

1. A zip wire runs between two posts, 25m apart. The zip wire is at an angle of 10^{\circ} to the horizontal. Calculate the length of the zip wire.

2. A surveyor wants to know the height of a skyscraper. He places his inclinometer on a tripod 1m from the ground. At a distance of 50m from the skyscraper, he records an angle of elevation of 82^{\circ} .

What is the height of the skyscraper? Give your answer to one decimal place.

Total height = 355.8+1=356.8m.

3. Triangle ABC is isosceles. Work out the height of triangle ABC.

To solve this we split the triangle into two right angled triangles.

KS3 trigonometry questions – missing angles

4. A builder is constructing a roof. The wood he is using for the sloped section of the roof is 4m long and the peak of the roof needs to be 2m high. What angle should the piece of wood make with the base of the roof?

5. A ladder is leaning against a wall. The ladder is 1.8m long and the bottom of the ladder is 0.5m from the base of the wall. To be considered safe, a ladder must form an angle of between 70^{\circ} and 80^{\circ} with the floor. Is this ladder safe?

Not enough information

Yes it is safe.

6. A helicopter flies 40km east followed by 105km south. On what bearing must the helicopter fly to return home directly?

Since bearings are measured clockwise from North, we need to do 360-21=339^{\circ}.

KS4 trigonometry questions

In KS4 maths, trigonometry questions ask students to solve a variety of problems including multi step problems and real life problems. We also need to be familiar with the exact values of the trigonometric functions at certain angles.

In the higher syllabus we look at applying trigonometry to 3D problems as well as using the sine rule, cosine rule and area of a triangle.

Trigonometry is covered by all exam boards, including Edexcel, AQA and OCR.

Read more: Question Level Analysis Of Edexcel Maths Past Papers (Foundation)

KS4 trigonometry questions – SOHCAHTOA

7.   Calculate the size of angle ABC. Give your answer to 3 significant figures.

8. Kevin’s garden is in the shape of an isosceles trapezium (the sloping sides are equal in length). Kevin wants to buy enough grass seed for his garden. Each box of grass seed covers 15m^2 . How many boxes of grass seed will Kevin need to buy?

To calculate the area of the trapezium, we first need to find the height. Since it is an isosceles trapezium, it is symmetrical and we can create a right angled triangle with a base of \frac{10-5}{2} .

We can then find the area of the trapezium:

Number of boxes: 88.215=5.88

Kevin will need 6 boxes.

KS4 trigonometry questions – exact values

9.   Which of these values cannot be the value of \sin(\theta) ?

10. . Write 4sin(60) + 3tan(60) in the form a\sqrt{k}.

KS4 trigonometry questions – 3D trigonometry

Work out angle a, between the line AG and the plane ADHE.

We need to begin by finding the length AH by looking at the triangle AEH and using pythagoras theorem.

\begin{aligned} &AH^2=14^2+3^2 \\\\ &AH^2=205 \\\\ &AH=14.32cm \end{aligned}

We can then find angle a by looking at the triangle AGH.

\begin{aligned} \tan(\theta)&=\frac{O}{A}\\\\ \tan(\theta)&=\frac{4}{14.32}\\\\ \theta&=tan^{-1}(\frac{4}{14.32})\\\\ \theta&=15.6^{\circ} \end{aligned}

12.   Work out the length of BC.

First we need to find the length DC by looking at triangle CDE.

We can then look at triangle BAC.

KS4 trigonometry questions – sine/cosine rule

13. Ship A sails 40km due West and ship B sails 65km on a bearing of 050^{\circ} . Find the distance between the two ships.

The angle between their two paths is 90+50=140^{\circ} .

\begin{aligned} a^{2}&=b^{2}+c^{2}-2bc \cos(A)\\\\ a^{2}&=40^{2}+65^{2}-2\times 40 \times 65 \cos(140)\\\\ a^{2}&=5825-5200 \cos(140)\\\\ a^{2}&=9808.43\\\\ a&=99.0\mathrm{km} \end{aligned}

14.   Find the size of angle B.

First we need to look at the right angled triangle.

Then we can look at the scalene triangle.

KS4 trigonometry questions – area of a triangle

The area of the triangle is 16cm^2 . Find the length of the side x .

\begin{aligned} \text{Area }&=\frac{1}{2}ab \sin(C)\\\\ 16&=\frac{1}{2} \times x \times 2x \times \sin(40)\\\\ 16&=x^{2} \sin(40)\\\\ \frac{1}{\sin(40)}&=x^{2}\\\\ 24.89&=x^{2}\\\\ 5.0&=x \end{aligned}

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Trigonometry : Solving Word Problems with Trigonometry

Study concepts, example questions & explanations for trigonometry, all trigonometry resources, example questions, example question #1 : solving word problems with trigonometry.

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

You can draw the following right triangle from the information given by the question.

In order to find the height of the flagpole, you will need to use tangent.

You can draw the following right triangle from the information given in the question:

In order to find out how far up the ladder goes, you will need to use sine.

In right triangle ABC, where angle A measures 90 degrees, side AB measures 15 and side AC measures 36, what is the length of side BC?

This triangle cannot exist.

Example Question #5 : Solving Word Problems With Trigonometry

A support wire is anchored 10 meters up from the base of a flagpole, and the wire makes a 25 o angle with the ground. How long is the wire, w? Round your answer to two decimal places.

23.81 meters

28.31 meters

21.83 meters

To make sense of the problem, start by drawing a diagram. Label the angle of elevation as 25 o , the height between the ground and where the wire hits the flagpole as 10 meters, and our unknown, the length of the wire, as w. 

Now, we just need to solve for w using the information given in the diagram. We need to ask ourselves which parts of a triangle 10 and w are relative to our known angle of 25 o . 10 is opposite this angle, and w is the hypotenuse. Now, ask yourself which trig function(s) relate opposite and hypotenuse. There are two correct options: sine and cosecant. Using sine is probably the most common, but both options are detailed below.

We know that sine of a given angle is equal to the opposite divided by the hypotenuse, and cosecant of an angle is equal to the hypotenuse divided by the opposite (just the reciprocal of the sine function). Therefore:

To solve this problem instead using the cosecant function, we would get:

The reason that we got 23.7 here and 23.81 above is due to differences in rounding in the middle of the problem. 

Example Question #6 : Solving Word Problems With Trigonometry

When the sun is 22 o above the horizon, how long is the shadow cast by a building that is 60 meters high?

To solve this problem, first set up a diagram that shows all of the info given in the problem. 

Next, we need to interpret which side length corresponds to the shadow of the building, which is what the problem is asking us to find. Is it the hypotenuse, or the base of the triangle? Think about when you look at a shadow. When you see a shadow, you are seeing it on something else, like the ground, the sidewalk, or another object. We see the shadow on the ground, which corresponds to the base of our triangle, so that is what we'll be solving for. We'll call this base b.

Therefore the shadow cast by the building is 150 meters long.

If you got one of the incorrect answers, you may have used sine or cosine instead of tangent, or you may have used the tangent function but inverted the fraction (adjacent over opposite instead of opposite over adjacent.)

Example Question #7 : Solving Word Problems With Trigonometry

From the top of a lighthouse that sits 105 meters above the sea, the angle of depression of a boat is 19 o . How far from the boat is the top of the lighthouse?

423.18 meters

318.18 meters

36.15 meters

110.53 meters

To solve this problem, we need to create a diagram, but in order to create that diagram, we need to understand the vocabulary that is being used in this question. The following diagram clarifies the difference between an angle of depression (an angle that looks downward; relevant to our problem) and the angle of elevation (an angle that looks upward; relevant to other problems, but not this specific one.) Imagine that the top of the blue altitude line is the top of the lighthouse, the green line labelled GroundHorizon is sea level, and point B is where the boat is.

Merging together the given info and this diagram, we know that the angle of depression is 19 o  and and the altitude (blue line) is 105 meters. While the blue line is drawn on the left hand side in the diagram, we can assume is it is the same as the right hand side. Next, we need to think of the trig function that relates the given angle, the given side, and the side we want to solve for. The altitude or blue line is opposite the known angle, and we want to find the distance between the boat (point B) and the top of the lighthouse. That means that we want to determine the length of the hypotenuse, or red line labelled SlantRange. The sine function relates opposite and hypotenuse, so we'll use that here. We get:

Example Question #8 : Solving Word Problems With Trigonometry

Angelina just got a new car, and she wants to ride it to the top of a mountain and visit a lookout point. If she drives 4000 meters along a road that is inclined 22 o to the horizontal, how high above her starting point is she when she arrives at the lookout?

9.37 meters

1480 meters

3708.74 meters

10677.87 meters

1616.1 meters

As with other trig problems, begin with a sketch of a diagram of the given and sought after information.

Angelina and her car start at the bottom left of the diagram. The road she is driving on is the hypotenuse of our triangle, and the angle of the road relative to flat ground is 22 o . Because we want to find the change in height (also called elevation), we want to determine the difference between her ending and starting heights, which is labelled x in the diagram. Next, consider which trig function relates together an angle and the sides opposite and hypotenuse relative to it; the correct one is sine. Then, set up:

Therefore the change in height between Angelina's starting and ending points is 1480 meters. 

Example Question #9 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 50 feet apart. The shorter building is 40 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 48 o . How high is the taller building?

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

Example Question #10 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 80 feet apart. The shorter building is 55 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 32 o . How high is the taller building?

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Category:Trigonometry Problems

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Graphs of Trigonometric Functions Practice Questions

Trigonometric graphs are perhaps one of the most basic and essential elements to grasp in relation to many different concepts and uses in mathematics. These are sine, cosine and tangent; entities that are used to describe periodic behavior of wave such as sound, light, mechanical etc. and even motion of pendulums. To the students, mastering such graphs is very important if they are going to tackle higher classes in trigonometry, calculus and the rest of them.

This article seeks to present a so-called users’ guide to graphs of the trigonometric functions, their characteristics and usage in the real world.

Basic Trigonometric Functions

There are six basic trigonometric functions that are discussed as follows:

Transformations of Trigonometric Functions

Some of the common transformations of trigonometric functions are:

• Vertical Stretching/Compressing
• Horizontal Stretching/Compressing

Phase Shifts (Horizontal Translations)

Vertical shifts, amplitude and vertical stretching/compressing.

The amplitude of a trigonometric function, is measured by the greatest distance that can be obtained from the middle of the wave to the crest of the wave. The amplitude is always denoted by ∣A∣ If the function contains a form y = A sin(x) or a form y = A cos(x).

Examples: The function y = 2sin(x) has the amplitude equal to 2, therefore, the graph of the sine function is stretched vertically by a factor of 2.

Period and Horizontal Stretching/Compressing

The range of a trigonometric function is the measure of a portion of the independent variable in going through a cycle. The period is 2π/ ∣B∣ for y=sin(Bx) or y=cos(Bx).

Examples: y = sin 2x has coefficient 2 in parenthesis, so it has a period of π, making the graph more condensed horizontally.

The graph of y = cos( x/2) is said to be a horizontal stretch of the graph of y = cos x by a factor of 2 and hence has a period of 4π.

A phase shift means shifting the whole graph horizontally in the x-direction. When y = sin(x − C) or y = cos(x − C), the graph is shifted right by C if C > 0 and left by |-C| if C < 0.

Examples: y = sin(x − π/2) makes the sine graph move 1/2 π units to the right.

y = cos(x + π/3) translates/Shifts the cosine graph to the left side by π/3 units.

Shift up of the entire graph is referred to as vertical shift. When y = sin(x) + D or y = cos(x) + D, then while, the graph is shifted D units up when D is positive or D unit down if D is negative.

Examples: y = sin(x) + 2 acts to move the sine graph up 2 units.

y = cos(x − 1) shifts the cosine graph 1 unit right, or we can equivalently say that it shifts the cosine graph 1 unit down.

Solved Problems

Problems 1. Sketch y = 2 sin(x) over the interval [0, 2π]

The graph varies between -2 and 2 and contains a period of 2π. Some of the ideal points include; (0,0), (∼ 1/4π, 2), (∼ π, 0), (∼ 3/4π, -2), and (∼ 2π, 0).

Problems 2. Determine the period and sketch y = sin (2x) over the interval [0, π]

The period of y = sin(2x) is π. Key points: (0,0) (π/4, 1) (π/2, 0) (3π/4, -1) (π, 0)

Problems 3. Determine the phase shift and sketch y = cos(x – π/4)

The graph of the given equation is shifted π/4 units to the right. , and more of them are (π/4, 1), (3π/4, 0), (5π/4, -1), (7π/ 4, 0), and (9π/4, 1).

Problems 4. Find the vertical shift and sketch y = tan(x) + 1

The graphs on the given set of drawings are shifted 1 unit up. It goes to infinity and has vertical asymptotes at x = π/2 + nπ and passes through the points (0,1), π/2 + nπ, etc.

Problems 5. Combine transformations and sketch y = 3 sin(2x − π/2) − 2.

The graph will have a maximum value of 3 and a minimum of -3, it will take π units to complete one cycle and starts at π/4 from the y-axis and it is also two units below the x-axis. Some other key points passing through the given hyperbolic paraboloid are (π/4,−2), (π/2,1), (3π/4,−2), (π,−5), and (5π/4,−2).

Problems 6. Determine the amplitude, period, and vertical shift for y = -1/2 cos(4x) + 1

Amplitude is 1/2, period is π/2, and vertical shift is 1. Key points: (0,1) (π/8, 0.5) (π/4, 1) (3π/8, 1.5) (π/2, 1)

Problems 7. Find the period and phase shift of y = tan(x + π/4)

The period is π, and the phase shift is π/4 to the left. Key points: Vertical asymptotes at x = -π/4 and x = π/4 Passes through points (-π/4, 0) and (0, 1).

Problems 8. Sketch y = cos(3x) over the interval [0, 2π]

The period of y = cos(3x) is 2π/3. Key points: (0, 1) (π/6, 0) (π/3, -1) (π/2, 0) (2π/3, 1)

Problems 9. Combine transformations and sketch y = 3 sin(x/2) – 2 over the interval [0, 4π]

The amplitude is 3, the period is 4π, and the vertical shift is -2. Key points: (0, -2) (2π, 1) (4π, -2) (6π, -5) (8π, -2)

Problems 10. Determine the vertical shift and sketch y = sin(x) + 3 over the interval [0, 2π]

The graph is shifted 3 units up. Key Points: (0,3) (π/2, 4) (π, 3) (3π/2, 2) (2π, 3)

Unsolved Practice Questions

Questions 1. Sketch y= sin(3x) over the interval [0,2π].

Questions 2. Determine the phase shift and sketch y = cos(x + π/2) over the interval [0. 2π].

Questions 3. Find the vertical shift and sketch y = tan(x) – 3 over the interval [0, 2π].

Questions 4. Combine transformations and sketch y = -sin(x/2) + 1 over the interval [0, 4π].

Questions 5. Sketch y= sec(x) over the interval [0, 2π].

Questions 6. Determine the amplitude, period, and phase shift for y = 4sin(2x – π).

Questions 7. Find the period and sketch y = cos(x – π/4) over the interval [0, 2π].

Questions 8. Combine transformations and sketch y = cos(4x) – 1 over the interval [0, π].

Questions 9. Sketch y = cot(x/2) over the interval [0, 2π].

Questions 10. Determine the vertical shift and sketch y = sec(x) + 2 over the interval [0, 2π].

Mastering the graphs of trigonometric functions, to be familiar with them and their usage in different areas, it is necessary to learn their graphs. The nice thing about identifying amplitude, period, phase shifts, and vertical shifts is that students are able to correctly graph and analyze these functions. This article is so written to give the students a blow-by-blow account of these concepts so that they can get it right each time. In the course of carrying out the aforementioned functions, the students will be able to employ these functions when writing academically and when writing in real life situations through practice and understanding.

• Trigonometric Graphs
• Trigonometric Functions
• Graphs of Inverse trigonometric Functions

FAQs on Graphs of Trigonometric Functions

What is the amplitude of a trigonometric function.

The amplitude is the maximum extent that a wave oscillates from its midpoint or midline to the highest or the lowest point of the wave. For the function y = A sin(x), the coefficient of the sine in the function denotes the amplitude and therefore the amplitude is ∣A∣.

How do you determine the period of a trigonometric function?

The period is half of the distance over which the function reoccurs. Hence, the period are 2π / |B|, for y = sin(Bx).

What are phase shifts in trigonometric functions?

Translations are a specific type of phase shift and involve shifting the graph of the function horizontally. Regarding y = sin(x − C), the graph will be shifted C units to the right.

How do vertical shifts affect the graph of a trigonometric function?

Vertical translations change the location of the graph up or down on the y-axis leaving the basic appearance fixed. If we have y = sin(x) + D then the graph is shifted ‘D’ units up.

What are the key features of the tangent function graph?

Tangent has a period of π and has vertical asymptotes because it is undefined at those values, and there are no relative maximum and minimum points in tangent.

Why are trigonometric functions important in real-world applications?

Cohort, sound and light waves, and mechanical oscillations, trigonometric functions are used in relation to probability and statistics, physics and engineering, and signal processing.

How can transformations of trigonometric functions be combined?

The shifts including amplitude, period, phase shift and vertical shift can be compounded in order to give variants of the original graph of the function.

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