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Learn Index · Trigonometry Grade 9–10

Inverse trigonometric ratios

sin⁻¹, cos⁻¹ and tan⁻¹ turn a ratio back into an angle.

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Going backwards

The trig ratios turn an angle into a ratio: sin 30° = 0.5.

The inverse trigonometric ratios turn a ratio back into an angle:

sin⁻¹(0.5) = 30°

Function Also called Asks
sin⁻¹ x arcsine "which angle has sine x?"
cos⁻¹ x arccosine "which angle has cosine x?"
tan⁻¹ x arctangent "which angle has tangent x?"

The −1 is not an exponent here. sin⁻¹ x is not 1/sin x.

Finding an angle in a right triangle

  1. Label the sides from the unknown angle.
  2. Choose the ratio for the two known sides.
  3. Apply the inverse function.

Example: opposite = 9, hypotenuse = 15

  • sin θ = 9/15 = 0.6
  • θ = sin⁻¹(0.6) ≈ 36.9°

Example: adjacent = 7, hypotenuse = 10

  • θ = cos⁻¹(0.7) ≈ 45.6°

Example: opposite = 11, adjacent = 4

  • θ = tan⁻¹(11/4) ≈ 70.0°

Calculator steps

On most scientific calculators:

  1. Make sure you’re in degree mode (DEG).
  2. Press SHIFT (or 2nd), then SIN, to get sin⁻¹.
  3. Enter the ratio and close the bracket.

Use the exact fraction (9 ÷ 15), not a rounded decimal, for accuracy.

Valid inputs

  • sin⁻¹ and cos⁻¹ only accept values from −1 to 1. sin⁻¹(1.4) gives an error, because no angle has a sine of 1.4. If you get this, check your side labels: the hypotenuse must be the longest side.
  • tan⁻¹ accepts any number.

Finding both acute angles

In a right triangle, the two acute angles add to 90°. Find one with an inverse ratio, then subtract.

Example: legs 5 and 12

  • θ = tan⁻¹(5/12) ≈ 22.6°
  • Other angle ≈ 90° − 22.6° = 67.4°

Exact values

From special triangles:

  • sin⁻¹(½) = 30°
  • cos⁻¹(√2/2) = 45°
  • tan⁻¹(√3) = 60°

Real life: ramps and roofs

A ramp rises 1 m over a run of 12 m. Angle of incline:

tan⁻¹(1/12) ≈ 4.8°

Worked examples

Example 1: cos θ = 0.25. θ? cos⁻¹(0.25) ≈ 75.5°.

Example 2: tan θ = 1. θ? 45°.

Example 3: A 6 m ladder reaches 5.5 m up a wall. Angle with the ground? sin⁻¹(5.5/6) ≈ 66.4°.

Common mistakes

  • Treating sin⁻¹ as 1/sin.
  • Calculator in radian mode.
  • Rounding the ratio too early.

Try it yourself

  1. sin θ = 0.8. Find θ.
  2. Opposite 6, adjacent 9. Find θ.
  3. Adjacent 12, hypotenuse 13. Find θ.
  4. Find both acute angles in a 7–24–25 triangle.

Answers: 1) ≈ 53.1° 2) ≈ 33.7° 3) ≈ 22.6° 4) ≈ 16.3° and ≈ 73.7°

Quick recap

  • Inverse trigonometric ratios find angles from ratios.
  • sin⁻¹ (arcsine), cos⁻¹, tan⁻¹; the −1 isn’t an exponent.
  • Use degree mode and exact ratios.