Trigonometry

Trigonometry is one of the subjects in mathematics that studies the ratios between sides of right-angled triangle.

Sine, Cosine, and Tangent Ratio

sin θ = Opposite / Hypotenuse, cos θ = Adjacent / Hypotenuse, tan θ = Opposite / Adjacent

SOH CAH TOA

Calculating Sides

1. Find the length of side a.

Right-angled triangle ABC with angle 43° at A, CA = 11 and BC = a

The information given is the adjacent side of the given angle, and the question is the length of opposite side, so we use tangent ratio.

\tan 43 ° = \frac{BC}{CA} = \frac{a}{11}

a = 11 \tan 43 °

a = 10.26

2. A boy is flying a kite at the end of a 10 m string. The angle of elevation of the kite from the boy is 32°. Find the height of the kite.

Angle of elevation problem: kite on a 10 m string at 32°, find height p m

The information given is the hypotenuse side of the given angle, and the question is the length of opposite side, so we use sine ratio.

\sin 32 ° = \frac{p}{10}

p = 10 \sin 32 °

p = 5.3 m

So, the height of kite is 5.3 m.

3. Find the length of side q correct to 3 significant figures.

Staircase 35 m long at 26° to the horizontal, horizontal length q m

The information given is the hypotenuse side of the given angle, and the question is the length of adjacent side, so we use cosine ratio.

\cos 26 ° = \frac{q}{35}

q = 35 \cos 26 °

q = 31.5

So, the value of q is 31.5.

Calculating Angles

1. An aeroplane takes off from point A at angle of elevation θ°. Calculate θ.

Aeroplane taking off from A: horizontal 400, vertical 300, angle θ

The information given is the length of opposite and the adjacent side, so we use tangent ratio.

\tan θ = \frac{300}{400} = \frac{3}{4}

θ = \tan^{- 1} (\frac{3}{4}) = 36.9 °

2. An isosceles triangle of sides 100 cm, 60 cm and 60 cm has a base angle θ. Find θ.

Isosceles triangle with sides 60 cm, 60 cm and base 100 cm split into two 50 cm halves by height h; base angle θ

Component right-angled triangle with hypotenuse 60 cm, base 50 cm, height h and angle θ

To find the value of θ, we can use cosine ratio,

\cos θ = \frac{50}{60} = \frac{5}{6}

θ = \cos^{- 1} (\frac{5}{6})

= 33.6 °

Exercise

A hawk sits on top of a tree and sees a mouse on the ground at an angle of depression 25°. The mouse is 25 m away from the base of the tree. Find the height of the tree.
A hot-air balloon rises vertically from the ground stopping h₁ m above the ground before rising h₂ m further.
1. Find the heights h₁ and h₂.
2. The balloon takes 1 minute to rise from the ground to h₁ m. It takes a further 5 mins to rise to its highest point. Calculate the average speed of the total ascent in m/s.
Zeno sails his boat 20 km from Port A on a bearing of 030° to point B. He then sails 10 km on bearing 120° to point C arriving at 3 pm. A storm warning tells him to return directly to Port A before 5:15 PM.
1. Draw a diagram of Zeno's route and use it to find the bearing of his return journey.
2. If distance CA = $10 \sqrt{5}$ km and he returns at 10 km/h, find out whether he returns to Port A before 5:15 PM.

Answer

1. This is the illustration of the problem.

A hawk on a tree sees a mouse on the ground 25 m away at an angle of depression θ

It is given that the angle of depression is 25°, then it is equal to the angle of elevation from the mouse to the hawk. The height of the tree can be found by using tangent ratio,

\tan 25 ° = \frac{h}{25}

h = 25 \tan 25 °

h = 11.7 m

So, the height of the tree is 11.7 m.

Past Paper Questions

Cambridge Mathematics IGCSE 0580/22 March 2026

Question 17: right-angled triangle ABC, angle B = 90°, angle C = 30°, BC = 12 cm, not to scale

(a) Find the length of AC.
Give your answer in the form $a \sqrt{b}$ . [4]

\cos 30 ° = \frac{BC}{AC}

\frac{1}{2} \sqrt{3} = \frac{12}{AC}

AC = \frac{12}{\frac{1}{2} \sqrt{3}} = \frac{24}{\sqrt{3}} = \frac{24 \sqrt{3}}{3} = 8 \sqrt{3}

(b) Find the shortest distance from B to AC. [3]

Triangle ABC with the perpendicular d drawn from B to AC

\sin 30 ° = \frac{d}{12}

\frac{1}{2} = \frac{d}{12}

d = 6