Sheet № 110 · Foundation + Higher · AQA · Edexcel · OCR
SOHCAHTOA Finding Missing Angles –
Finding a missing angle is the reverse of finding a missing side. Instead of using sin, cos or tan directly, you use their inverse functions to work backwards from a ratio to an angle.
§Key definitions
Question:
A right-angled triangle has an opposite side of 6 cm and a hypotenuse of 10 cm. Find the angle θ. Give your answer to 1 decimal place.
Answer:
θ = 36.9° (1 d.p.).
Q1 (Foundation):
A right-angled triangle has O = 5 cm and H = 13 cm. Find angle θ to 1 d.p.
Q2 (Foundation):
A right-angled triangle has A = 12 cm and H = 15 cm. Find angle θ to 1 d.p.
Q3 (Higher):
From the top of a 50 m building, the angle of depression to a car is measured. The car is 120 m from the base of the building. Find the angle of depression to 1 d.p.
§Formulas to memorise
θ = sin⁻¹(Opposite ÷ Hypotenuse)
θ = cos⁻¹(Adjacent ÷ Hypotenuse)
θ = tan⁻¹(Opposite ÷ Adjacent)
The ladder is the hypotenuse (H = 5). The distance from the wall is the adjacent side to the angle at the ground (A = 1.5).
Worked example
A right-angled triangle has an opposite side of 6 cm and a hypotenuse of 10 cm. Find the angle θ. Give your answer to 1 decimal place.
Working: Label: O = 6, H = 10. O and H are involved, so use sin. sin θ = 6 ÷ 10 = 0.6 θ = sin⁻¹(0.6)
⚠ Common mistakes
- ✗Forgetting to use the inverse function. If tan θ = 1.875, the angle is tan⁻¹(1.875), not just 1.875. Students who write 1.875° as the answer lose all accuracy marks.
- ✗Calculator in radian mode. If your answer seems unusually small (like 0.6 instead of 36.9°), check the mode. The display should show D or DEG.
- ✗Dividing the wrong way round. For tan, the opposite goes on top and the adjacent on the bottom. Swapping them gives the wrong angle (you would get the complementary angle instead).
✦ Exam tips
- →Write the ratio and the fraction before using the inverse function — this earns a method mark.
- →Know the common exact values: sin⁻¹(0.5) = 30°, cos⁻¹(0.5) = 60°, tan⁻¹(1) = 45°.
- →After finding one angle, you can find the other non-right angle using the fact that angles in a triangle sum to 180°.
- →If a question involves elevation or depression, sketch the right-angled triangle and label the angle carefully — the angle of depression from the top equals the angle of elevation from the bottom (alternate angles).