Exact trigonometric values

In a nutshell

While most trigonometric values are very hard (or downright impossible) to find without using a calculator, there are very specific values that can be memorised.

Trigonometry values to memorise

$x$	$\sin(x)$	$\cos(x)$	$\tan(x)$
$0$	$0$	$1$	$0$
$30$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$45$	$\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$	$\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$	$1$
$60$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90$	$1$	$0$	$n/a$

An easier way to learn the values

This may seem like a lot to memorise, but there are easier ways to remember these. The table below is easy to construct when you realise the numbers are consecutive ( $0,1,2,3,4$ ). To use this table:

Maths; Trigonometry; KS4 Year 10; Exact trigonometric values

Procedure

1.	Locate the number associated with the trigonometric function and angle you want.
2.	Take the number, square root it, and then divide by 2 (hence the design of the table).

Example 1

What is the value of $\cos(60)$ ?

Refer to the table to see that $\cos(60)$ corresponds to the number 1.

Maths; Trigonometry; KS4 Year 10; Exact trigonometric values

Hence, square root this value, and then halve it:

$\underline{\cos(60) = \frac{\sqrt{1}}{2} = \frac{1}{2}}$

What about $\tan()$ ?

The table only gives values for $\sin()$ and $\cos()$ . The easiest way to learn values for $\tan()$ is to just memorise the four values. Alternatively, you can find the value for $\sin()$ and $\cos()$ of the particular angle and divide them.

Example 2

What is the value of $\tan(30)$ ?

First find $\sin(30)$ by referring to the table. The value in the table is $1$ , giving

$\sin(30)= \dfrac 1 2$ 

Then find $\cos(30)$ , using the same method, giving

$\cos(30)=\dfrac {\sqrt3}{2}$ 

Find the value of $\tan(30)$ by dividing $\sin(30)$ by $\cos(30)$

$\tan(30) = \dfrac{\sin(30)}{\cos(30)} = \dfrac{\frac 1 2}{\frac {\sqrt3}{2}} = \dfrac {1}{\sqrt3}=\underline{\dfrac{\sqrt3}{3}}$ 

Common trigonometric value problems

You should know how to apply the common trigonometric values to standard trigonometry with a right-angled triangle.

Example 3

Find the exact value of $x$ in the diagram below.

Maths; Trigonometry; KS4 Year 10; Exact trigonometric values

To solve this, treat this like a normal trigonometric question.

Label the diagram:

Maths; Trigonometry; KS4 Year 10; Exact trigonometric values

The adjacent and hypotenuse are the important lengths here. So, use CAH:

$\cos(x)=\frac{A}{H}$

Substitute in values:

$\cos(45)=\frac{10}{x}$

Rearrange to solve for the hypotenuse:

$x=\frac{10}{\cos(45)}$ 

Use the common trigonometric ratios to find the value of $\cos(45)$ :

$\cos(45)=\frac{1}{\sqrt{2}}$

Substitute this value in to find the exact value of $x$ :

$x=\frac{10}{\frac{1}{\sqrt{2}}}$

$\underline{x=10\sqrt{2}cm}$

Note: When a question asks for you to find the exact value, it means that you should leave your answer as a surd or fraction.