Exact Trig Values (Cambridge (CIE) IGCSE Maths: Extended): Revision Note

Exact trig values

What are exact values in trigonometry?

• For certain angles the values of sin θ, cos θ and tan θ can be written exactly

  • This means using fractions and surds

• You are expected to know the exact values of sin, cos and tan for

  • 0°, 30°, 45°, 60°, 90°, 180° and their multiples

How can I remember these exact trig values?

• Look at patterns in the table

  • Note the values of sin θ from 0° to 90° match cos θ in reverse, from 90° to 0°

• Two special right-angled triangles below can help you to find the exact values for 30º, 45º and 60º

  • Remember that by rationalising the denominator, $\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

Triangle for 45°

Triangle for 30° and 60°

• The trig graphs can help you to remember the exact values for 0º and multiples of 90º

How do I use exact trig values?

• You may come across trig questions in a non-calculator question

• In trig calculations, substitute in the exact trig values and solve as usual

  • E.g. Solve the equation $\cos 45=\frac{x}{12}$

    • Replace $\cos 45$ with $\frac{\sqrt{2}}{2}$ to give $\frac{\sqrt{2}}{2}=\frac{x}{12}$

    • Then you can solve for $x$

• On trig graphs, you may be expected to find a coordinate

  • E.g. The coordinates $\left(30,k\right)$ lie on the graph $y=\tan x$, find $k$

    • $k$ will be equal to $\tan 30$

    • The exact value of $\tan 30$ is $\frac{\sqrt{3}}{3}$

    • Therefore $k=\frac{\sqrt{3}}{3}$