There are some questions in Higher GCSE exam papers where you need to calculate lengths of triangles using trigonometry without a calculator.
The only way to solve these types of question is to learn exact trig values.
The values of sin, cos and tan that you need to know are those for 0°, 30°, 45° 60° and 90°.
Deriving the values
Take an equilateral triangle, giving each side a unit of 2:
Now cut it in half to make a right-angled triangle as shown:
Using Pythagoras, we can deduce that the vertical side is √3.
We now have a right-angled triangle with three known lengths and angles. It means we can apply sin, cos, and tan to work out their values for 30° and 60° by considering opposite, adjacent, and hypotenuse sides and using SOH CAH TOA
sin 60 = √3/2
cos 60 = 1/2
tan 60 = √3/1
sin 30 = 1/2
cos 30 = √3/2
tan 30 = 1/√3
Now, consider a right-angled isosceles triangle as shown:
We can give the two shorter lengths a unit of 1, and using Pythagoras, deduce the hypotenuse is √2.
Once again, we can use SOH CAH TOA, this time to work out all the values for 45°:
sin 45 = 1/√2
cos 45 = 1/√2
tan 45 = 1/1
Learning the table
The table below summarises all the values you need to learn.
Each value has been simplified and rationalised.
As well as the values we calculated from our triangles above, you also need to be familiar with sin, cos, and tan of 0° and 90°.
Tips for remembering the table
The columns for sin and cos are the same list of values, but in reverse order.
cos is the only trig ratio that doesn't have a value of zero at 0°.
tan x = sin x / cos x. This means you can derive any value for tan by dividing sin and cos of the same angle. (e.g. tan 60 = sin 60 / cos 60 = √3/2 ÷ 1/2 = √3)
tan 90 has no value because it would require us to divide 1 by 0, which is infinity.
This is a tricky activity but it will become easier if you practise doing some questions.