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.

What Is Finding Missing Angles with SOHCAHTOA?

When you know two sides of a right-angled triangle and need to find an angle, you use the inverse trigonometric functions. These are written as sin⁻¹ (or arcsin), cos⁻¹ (or arccos), and tan⁻¹ (or arctan) on your calculator, often accessed by pressing SHIFT then sin, cos or tan.

The process is similar to finding a side: you label O, A, H relative to the angle you want, choose the correct ratio based on which two sides you know, form the fraction, and then use the inverse function to convert that fraction into an angle.

It is essential that your calculator is set to degree mode (not radians) for GCSE. Common angle values worth recognising include sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1, sin 45° = cos 45° = √2/2, and sin 60° = cos 30° = √3/2.

Key Formulas

Step-by-Step Method

1. Label the two known sides as O, A, or H relative to the unknown angle.
2. Choose the ratio that uses those two sides (SOH, CAH, or TOA).
3. Form the fraction (e.g. O ÷ H).
4. Use the inverse function on your calculator (e.g. sin⁻¹).
5. Round as instructed by the question.

Worked Example 1 — Foundation Level

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.

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)

Answer: θ = 36.9° (1 d.p.).

Worked Example 2 — Higher Level

Question: A right-angled triangle has an adjacent side of 8 cm and an opposite side of 15 cm. Find the angle θ to 1 decimal place.

Working: Label: O = 15, A = 8. O and A are involved, so use tan. tan θ = 15 ÷ 8 = 1.875 θ = tan⁻¹(1.875)

Answer: θ = 61.9° (1 d.p.).

Worked Example 3 — Exam Style

Question: A 5 m ladder leans against a wall. The base of the ladder is 1.5 m from the wall. Find the angle the ladder makes with the ground. Give your answer to 1 decimal place.

Working: 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). cos θ = 1.5 ÷ 5 = 0.3 θ = cos⁻¹(0.3)

Answer: θ = 72.5° (1 d.p.).

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).

Practice Questions

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.

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Related Topics

• Trigonometry SOHCAHTOA
• SOHCAHTOA Finding Missing Sides
• Pythagoras' Theorem

Summary

• To find a missing angle, use the inverse trig function: sin⁻¹, cos⁻¹, or tan⁻¹.
• Label the two known sides as O, A, or H, choose the matching ratio, form the fraction, and press SHIFT + the trig button.
• Always check your calculator is in degree mode — a tiny answer is a sure sign it is in radians.
• Know the common exact values (sin 30° = 0.5, tan 45° = 1, cos 60° = 0.5) for Higher tier questions.
• Once you have found one angle, the other non-right angle is simply 180° minus 90° minus your answer.