Exact Trigonometric Values
Grade 5Why exact values matter
Some GCSE questions say "without using a calculator" or award marks for an exact answer like √3/2. You need to know the trig values for five key angles by heart.
The values to know
sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.
tan 0° = 0, tan 30° = 1/√3 (= √3/3), tan 45° = 1, tan 60° = √3, tan 90° = undefined.
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.
