Some angles have exact trig values you should know without a calculator. sin 30° = 1/2, cos 60° = 1/2, sin 45° = cos 45° = √2/2, tan 45° = 1, sin 60° = cos 30° = √3/2 and tan 30° = √3/3. Also sin 0° = 0, cos 0° = 1 and sin 90° = 1.
These come from two triangles: half a square (a 45°–45°–90° triangle with sides 1, 1, √2) and half an equilateral triangle (a 30°–60°–90° triangle with sides 1, √3, 2). Sketch them to rebuild any value you forget.
Worked example
Find the exact value of cos 30°.
- Use the half-equilateral triangle with sides 1, √3 and 2.
- Adjacent to 30° is √3 and the hypotenuse is 2.
Answer:√3/2
Key words
- exact value
- written with fractions and roots, not decimals
- surd
- a root such as √3
- equilateral triangle
- all sides and angles equal
Practise this
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