Exact Trigonometric Values

 

What are Exact Trigonometric Values?

For certain angles, the values of $\sin \theta$, $\cos \theta$, and $\tan \theta$ can be expressed exactly using fractions and surds.

These angles include: $0 \degree$, $30\degree$, $45\degree$, $60\degree$, $90\degree$, and their multiples.

Instead of using a calculator, you must memorise or derive these values using geometric reasoning.

 
Exact Trigonometric Values Table

$\theta$ (Degrees)	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0\degree$	0	1	0
$30\degree$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45\degree$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
$60\degree$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90\degree$	1	0	Undefined

 
How to Remember Exact Values?

1. Recognizing Patterns

• The sin values from $0\degree$ to $90\degree$ mirror the cos values in reverse from $90\degree$ to $0\degree$.
• tan values can be derived by using $\tan \theta = \sin \theta / \cos \theta$.

2. Using Special Right-Angled Triangles

Two well-known triangles help derive these values:

1. The $45\degree$-$45\degree$-$90\degree$ Triangle

• An isosceles right triangle with legs 1 unit and a hypotenuse $\sqrt{2}$.
• Gives values for $\sin 45\degree$, $\cos 45\degree$, and $\tan 45\degree$.

 

 

2. The $30\degree$-$60\degree$-$90\degree$ Triangle

• A right triangle where the shortest side is 1, the hypotenuse is 2, and the other side is $\sqrt{3}$.
• Helps derive $\sin 30\degree$, $\cos 30\degree$, $\sin 60\degree$, and $\cos 60\degree$.

 

 

Using Exact Trigonometric Values

In non-calculator exams, substitute exact values into equations instead of using decimal approximations.

 

Example 1: Solving an Equation

Find $x$ in the equation:

$$\cos 45\degree = \frac{x}{12}$$

Substituting $\cos 45\degree = \frac{\sqrt{2}}{2}$

$$\frac{\sqrt{2}}{2} = \frac{x}{12}$$

Rearrange:

$$x = 12 \times \frac{\sqrt{2}}{2}$$

$$x = 6\sqrt{2}$$

Final Answer: $x = 6\sqrt{2} cm$

 

Example 2: Finding a Coordinate on a Graph

A point $(30, k)$ lies on the graph of $y = \tan x$. Find $k$.

Since $k = \tan 30\degree$, substitute the exact value:
$$k = \frac{\sqrt{3}}{3}$$

Final Answer: $k = \frac{\sqrt{3}}{3}$