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).
Worked example
Try this one
Past paper - 2023
Taylor designs a logo using isosceles triangles joined at a central point, P.
This is the start of Taylor’s design.
The completed design will have rotational symmetry, order 60 about point P
Each triangle has base, b, and height, h, measured in mm.
Calculate h when b = 40mm.
Give your answer correct to 1 decimal place.
(a)----------------------------mm [4]
This question uses trigonometry inside an isosceles triangle. Because the design has rotational symmetry of a certain order, you can find the angle at point P by dividing 360° by that order.
Ask yourself: 'If I drop a perpendicular from P to the base, what right-angled triangle is formed, and which trig ratio connects the half-base and the height?'
For example, if a design had order 4, the central angle would be 90°, giving a half-angle of 45° — you would then use tan(45°) = (half base) ÷ height.
