Right-angled trigonometry

Trigonometry links the angles and sides of a right-angled triangle. First label the sides from the angle you're using (θ):

• Hypotenuse (H): the longest side, opposite the right angle
• Opposite (O): the side opposite θ
• Adjacent (A): the side next to θ that isn't the hypotenuse.

Example: triangle ABC has a right angle at B. Label its sides from the angle at A.

1. Opposite the right angle: AC is the hypotenuse
2. Opposite angle A: BC is the opposite
3. Next to angle A: AB is the adjacent

sin θ = O/H, cos θ = A/H, tan θ = O/A (on the formulae sheet). Remember them as SOH CAH TOA.

1. Label the sides, and tick the two you know or want.
2. Pick the ratio that uses those two.
3. Substitute and rearrange.

Example: the hypotenuse is 10 cm and the angle is 35°. Find the opposite side, to 1 d.p.

1. O and H: use sin
2. sin 35° = O/10, so O = 10 × sin 35° = 5.73…
3. 5.7 cm

Example: the adjacent side is 8 cm and the angle is 40°. Find the hypotenuse, to 1 d.p.

1. A and H: use cos
2. cos 40° = 8/H, so H = 8 ÷ cos 40° = 10.44…
3. 10.4 cm

To find an angle, use the inverse function: sin⁻¹, cos⁻¹ or tan⁻¹ (SHIFT then sin, cos or tan).

1. Label the two sides you know and pick the ratio.
2. Work out the fraction.
3. Use the inverse function to find the angle.

Example: the opposite side is 7 cm and the adjacent side is 9 cm. Find the angle, to 1 d.p.

1. O and A: use tan
2. tan θ = 7/9
3. θ = tan⁻¹(7/9) = 37.87…
4. 37.9°

• An angle of elevation is measured up from the horizontal.
• An angle of depression is measured down from the horizontal.

Example: you stand 30 m from the foot of a tree. The angle of elevation to its top is 28°. How tall is the tree, to 1 d.p.?

1. Adjacent = 30 m, and you want the opposite: use tan
2. Height = 30 × tan 28° = 15.95…
3. 16.0 m

Example: from the top of a 50 m cliff, the angle of depression to a boat is 12°. How far is the boat from the foot of the cliff, to 3 s.f.?

1. The angle at the boat is also 12° (alternate angles)
2. tan 12° = 50/d, so d = 50 ÷ tan 12° = 235.2…
3. 235 m

Learn these values: you need them without a calculator.

tan 90° has no value. √3/3 is the same as 1/√3.

Example: without a calculator, find the opposite side when the hypotenuse is 12 cm and the angle is 30°.

1. O = 12 × sin 30° = 12 × ½
2. 6 cm