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

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.

Set of 25 printable GCSE maths questions

Try these challenging GCSE maths questions with your students in class. They all include answers and are organised by difficulty!


How to answer trigonometry questions

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


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


\[\begin{aligned} \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \hspace{2cm} \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \hspace{2cm} \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \end{aligned}\]


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).

trigonometry questions step 2

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

trigonometry questions step 3


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.

\[\begin{aligned} &\text{Sine rule: }\; \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)} \\\\ &\text{Cosine rule: }\; a^{2}=b^{2}+c^{2}-2bc \cos(A)\\\\ &\text{Area of a triangle: }\; Area = \frac{1}{2}ab \sin(C) \end{aligned}\]


To answer the trigonometry question:

  1. Establish that it is not a right angled triangle.
  2. Label the sides of the triangle using lower case a, b, c.
  3. Label the angles of the triangle using upper case A, B and C.
  4. Opposite sides and angles should use the same letter so for example angle C is opposite to side c.
trigonometry questions step 4


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.


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.

 

trigonometry questions ks3 question 1

25.4m
GCSE Quiz True

144.0m
GCSE Quiz False

141.8m
GCSE Quiz False

24.6m
GCSE Quiz False

trigonometry questions ks3 question 1 answer

 

\begin{aligned} H&=\frac{A}{\cos(\theta)}\\\\ H&=\frac{25}{\cos(10)}\\\\ H&=25.4\mathrm{m} \end{aligned}

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.

 

trigonometry questions ks3 question 2

355.8m
GCSE Quiz False

7.0m
GCSE Quiz False

356.8m
GCSE Quiz True

49.5m
GCSE Quiz False

trigonometry questions ks3 question 2 answer

 

\begin{aligned} O&=\tan(\theta) \times A\\\\ O&=\tan(82) \times 50\\\\ O&=355.8\mathrm{m} \end{aligned}

 

Total height = 355.8+1=356.8m.

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

 

trigonometry questions ks3 question 3

6cm
GCSE Quiz False

17.4cm
GCSE Quiz True

34.9cm
GCSE Quiz False

2.1cm
GCSE Quiz False

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

 

trigonometry questions ks3 question 3 answer

 

\begin{aligned} O&=\tan(\theta) \times A\\\\ O&=\tan(71) \times 6\\\\ O&=17.4\mathrm{cm} \end{aligned}

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?

 

trigonometry questions ks3 question 4

26.6^{\circ}
GCSE Quiz False

60^{\circ}
GCSE Quiz False

0.008^{\circ}
GCSE Quiz False

30^{\circ}
GCSE Quiz True

 

trigonometry questions ks3 question 4 answer

 

\begin{aligned} \sin(\theta)&=\frac{O}{H}\\\\ \sin(\theta)&=\frac{2}{4}\\\\ \theta&=sin^{-1}(\frac{2}{4})\\\\ \theta&=30^{\circ} \end{aligned}

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?

 

 

trigonometry questions ks3 question 5

Yes

GCSE Quiz True

 No

GCSE Quiz False

Not enough information

GCSE Quiz False

trigonometry questions ks3 question 5 answer

 

\begin{aligned} \cos(\theta)&=\frac{A}{H}\\\\ \cos(\theta)&=\frac{0.5}{1.8}\\\\ \theta&=cos^{-1}(\frac{0.5}{1.8})\\\\ \theta&=73.9^{\circ} \end{aligned}

 

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?

21^{\circ}
GCSE Quiz False

 201^{\circ}
GCSE Quiz False

159^{\circ}
GCSE Quiz False

339^{\circ}
GCSE Quiz True

trigonometry questions ks3 question 6

 

\begin{aligned} \tan(\theta)&=\frac{O}{A}\\\\ \tan(\theta)&=\frac{40}{105}\\\\ \theta&=tan^{-1}(\frac{40}{105})\\\\ \theta&=21^{\circ} \end{aligned}

 

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)

A lesson introducing GCSE students to trigonometry using SOHCAHTOA on Third Space Learning's online intervention.
A lesson introducing GCSE students to trigonometry using SOHCAHTOA on Third Space Learning’s online intervention.


KS4 trigonometry questions – SOHCAHTOA

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

 

trigonometry questions ks4 question 7

24.6^{\circ}
GCSE Quiz False

38.4^{\circ}
GCSE Quiz True

18.8^{\circ}
GCSE Quiz False

21.8^{\circ}
GCSE Quiz False

trigonometry questions ks4 question 7 answer 1

 

\begin{aligned} A&=\frac{O}{\tan(\theta)}\\\\ A&=\frac{10}{\tan(28)}\\\\ A&=18.81\mathrm{m} \end{aligned}

 

trigonometry questions ks4 question 7 answer 2

 

\begin{aligned} \cos(\theta)&=\frac{A}{H}\\\\ \cos(\theta)&=\frac{18.81}{24}\\\\ \theta&=cos^{-1}(\frac{18.81}{24})\\\\ \theta&=38.4^{\circ} \end{aligned}

 

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?

 

trigonometry questions ks4 question 8

6
GCSE Quiz True

4
GCSE Quiz False

5
GCSE Quiz False

10
GCSE Quiz False

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} .

 

trigonometry questions ks4 question 8 answer

 

\begin{aligned} O&=\tan(\theta)\times A\\\\ O&=\tan(78)\times2.5\\\\ O&=11.76\mathrm{m} \end{aligned}

 

We can then find the area of the trapezium:

 

\begin{aligned} \text{Area }&=\frac{1}{2}(a+b)h\\\\ \text{Area }&=\frac{1}{2}(5+10)\times 11.76\\\\ \text{Area }&=88.2\mathrm{m}^{2} \end{aligned}

 

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) ?

\frac{1}{2}
GCSE Quiz False

\frac{\sqrt{3}}{2}
GCSE Quiz False

\frac{5}{2}
GCSE Quiz True

\frac{-\sqrt{2}}{2}
GCSE Quiz False
\sin\theta \text{ cannot be } \frac{5}{2} \text{ because } \sin\theta \text{ is always between 1 and -1}

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

5\sqrt{3}
GCSE Quiz True

2+3\sqrt{3}
GCSE Quiz False

\frac{3}{2} \sqrt{3}
GCSE Quiz False

\frac{1}{2}\sqrt{12} +3
GCSE Quiz False
\sin(60)=\frac{\sqrt{3}}{2}, \tan(60)=\sqrt{3}

 

\begin{aligned} 4\sin(60)+3\tan(60) &= 4\times \frac{\sqrt{3}}{2}+3\sqrt{3}\\\\ &=2\sqrt{3}+3\sqrt{3}\\\\ &=5\sqrt{3} \end{aligned}

KS4 trigonometry questions – 3D trigonometry

11.

trigonometry questions ks4 question 11

 

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

14.3^{\circ}
GCSE Quiz False

15.6^{\circ}
GCSE Quiz True

15.9^{\circ}
GCSE Quiz True

90^{\circ}
GCSE Quiz False

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

 

trigonometry questions ks4 question 11 step 1

 

\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.

 

trigonometry questions ks4 question 11 answer step 2

 

\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.

 

trigonometry questions ks4 question 12

30.0mm
GCSE Quiz False

19.0mm
GCSE Quiz False

23.3mm
GCSE Quiz False

29.6mm
GCSE Quiz True

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

 

trigonometry questions ks4 question 12 step 1

 

\begin{aligned} H&=\frac{O}{\sin(\theta)}\\\\ H&=\frac{15}{\sin(52)}\\\\ H&=19.04\mathrm{cm} \end{aligned}

 

We can then look at triangle BAC.

 

trigonometry questions ks4 question 12 step 2

 

\begin{aligned} H&=\frac{o}{\sin(\theta)}\\\\ H&=\frac{19.04}{\sin(40)}\\\\ H&=29.6\mathrm{mm} \end{aligned}

KS4 trigonometry questions – sine/cosine rule

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

 

trigonometry questions ks4 question 13

76.3km
GCSE Quiz False

99.0km
GCSE Quiz True

52.5km
GCSE Quiz False

84.9km
GCSE Quiz False

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

 

 

trigonometry questions ks4 question 13 answer

 

\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.

 

 

trigonometry questions ks4 question 14

78^{\circ}
GCSE Quiz False

31.8^{\circ}
GCSE Quiz False

9.3^{\circ}
GCSE Quiz False

28.8^{\circ}
GCSE Quiz True

First we need to look at the right angled triangle.

 

trigonometry questions ks4 question 14 step 1

 

\begin{aligned} &O=tan()A\\\\ &O=tan(22)23\\\\ &O=9.29cm \end{aligned}

 

Then we can look at the scalene triangle.

 

trigonometry questions ks4 question 14 step 2

 

\begin{aligned} \frac{\sin(A)}{A}&=\frac{\sin(B)}{B}\\\\ \frac{\sin(51)}{15}&=\frac{\sin(B)}{9.29}\\\\ 9.29\times \frac{\sin(51)}{15} &= \sin(B)\\\\ 0.481&=\sin(B)\\\\ \sin^{-1}(0.481)&=B\\\\ 28.8^{\circ}&=B \end{aligned}

KS4 trigonometry questions – area of a triangle

15.

trigonometry questions ks4 question 15

 

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

5.0cm
GCSE Quiz True

10.0cm
GCSE Quiz False

5.3cm
GCSE Quiz False

4cm
GCSE Quiz False

\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}

Looking for more keyword questions and resources?

Third Space Learning’s free GCSE maths resource library contains detailed lessons with step-by-step instructions on how to solve ratio problems, as well as worksheets with trigonometry practice questions and more GCSE exam questions.

Take a look at the trigonometry lessons today – more are added every week.

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Beki Christian
Beki Christian
Maths Teacher and Author
Beki has 11 years experience teaching secondary Maths. She is currently working as a Maths tutor as well as writing resources and blog posts for Third Space Learning.
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Downloadable resource

Set of 25 printable GCSE maths questions

Try these challenging GCSE maths questions with your students in class. They all include answers and are organised by difficulty!

Download Free Now!

Set of 25 printable GCSE maths questions

Downloadable resource

Try these challenging GCSE maths questions with your students in class. They all include answers and are organised by difficulty!

Download Free Now!