15 Trigonometry Questions And Practice Problems To Do With High Schoolers

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

Here we’ve provided 15 trigonometry questions that will help your students practice the various types of trigonometry questions they will encounter during high school.

Trigonometry in the real world

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

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Trigonometry in high school

In trigonometry we learn about the sine function, tangent function, and cosine function. These trig functions are abbreviated as sin, cos, and tan. We can use these to calculate sides and angles in right angled triangles. Later, students will be applying this to a variety of situations as well as learning the exact values of sin, cos, and tan for certain angles.

Students learn about trigonometric ratios: the law of sines, law of cosines, a new formula for the area of a triangle and applying trigonometric theorems to 3D shapes.

Trigonometry for more senior high school students will introduce the reciprocal trig functions, cotangent, secant and cosecant, but you don’t have to worry about these right now!


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

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse 

tan θ = opposite/adjacent

The acronym SOH CAH TOA is used so that you can remember which ratio to use.

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 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.

Sine rule: \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}

Cosine rule: a^{2}=b^{2}+c^{2}-2bc \cos(A)

Area of a triangle: Area = \frac{1}{2}ab \sin(C)

To answer the trigonometry question:

  1. Establish that it is not a right angled triangle.
  2. Label the sides of the triangle using lowercase 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, for example, angle A is opposite to side a.
trigonometry questions step 4


Trigonometry questions

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

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. Calculate 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}

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?

 

trigonometry questions ks3 question 6

 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}.

In geometry, 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. 

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

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 trapezoid (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 trapezoid, 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 trapezoid:

 

\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.

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}

Trigonometry questions: 3D trigonometry

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

 

trigonometry questions ks4 question 11

 

14.3^{\circ}
GCSE Quiz False

15.6^{\circ}
GCSE Quiz True

15.9^{\circ}
GCSE Quiz False

90^{\circ}
GCSE Quiz False

We need to begin by finding the length AH by looking at the triangle AEH and using pythagorean 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 False

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}

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.

 

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}

Trigonometry questions – area of a triangle

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

 

trigonometry questions ks4 question 15

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 high school trigonometry math questions and word problems?

Try these:

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The content in this article was originally written by secondary school maths teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.

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