Find an angle using tangent
Objective. Use inverse tangent to calculate an angle.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Finding an angle using tangent requires knowing both the opposite and adjacent sides of a right-angled triangle, then applying the inverse tangent function, written tan⁻¹, to the ratio of those two sides. The method is to calculate opposite ÷ adjacent first, then use a calculator's tan⁻¹ function on that decimal value to find the angle in degrees. For example, if the opposite side is 7 cm and the adjacent side is 9 cm, the angle is tan⁻¹(7 ÷ 9) ≈ tan⁻¹(0.778) ≈ 37.9°. This inverse operation reverses the usual tangent calculation, going from a ratio back to the angle that produced it.
Worked example
A right-angled triangle has an opposite side of 6 cm and an adjacent side of 10 cm. Find the angle using inverse tangent.
- 1Calculate the ratio opposite ÷ adjacent: 6 ÷ 10 = 0.6.
- 2Apply the inverse tangent function to this ratio: tan⁻¹(0.6).
- 3Using a calculator, tan⁻¹(0.6) ≈ 30.96°.
- 4Round to one decimal place: 31.0°.
Answer: Approximately 31.0°
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Practice
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Start practisingCommon mistake. Pupils sometimes apply tan⁻¹ to the adjacent-over-opposite ratio instead of opposite-over-adjacent, producing the complementary angle instead of the one required. Always divide opposite by adjacent before applying tan⁻¹.
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