Finding Angles with Trigonometry
Grade 5Going backwards from a ratio
When you know two sides of a right-angled triangle and want the angle, you use the inverse trig functions: sin⁻¹, cos⁻¹, or tan⁻¹ (also written arcsin, arccos, arctan).
They undo the trig function. If sin θ = 0.6, then θ = sin⁻¹(0.6). On your calculator this is usually the SHIFT or 2nd key followed by sin/cos/tan.
The method
1. Label the sides relative to the unknown angle (opposite, adjacent, hypotenuse).
2. Choose the correct ratio based on which two sides you know.
3. Write the equation (e.g. tan θ = 4/3).
4. Apply the inverse trig function: θ = tan⁻¹(4/3).
Make sure your calculator is in DEGREE mode (DEG), not RAD. A quick check: sin 30° should give exactly 0.5. If it doesn't, switch mode.
Worked example
EXAMPLE
A right-angled triangle has the side opposite angle θ = 7 cm and the adjacent side = 10 cm. Find θ.
tan θ = 7 ÷ 10 = 0.7. θ = tan⁻¹(0.7) = 35.0° (to 1 d.p.).
