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Exact trigonometric values

Tutor: Bilal

# 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

#### ​$\tan(x)$​​

$0$​​

$0$​​

$1$​​

$0$​​

$30$​​

$\frac{1}{2}$​​

$\frac{\sqrt{3}}{2}$​​

$\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$​​

$45$​​

$\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$​​

$\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$​​

$1$​​

$60$​​

$\frac{\sqrt{3}}{2}$​​​

$\frac{1}{2}$​​​

$\sqrt{3}$​​

$90$​​

​​$1$​​

$0$​​

$n/a$

## 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:

#### 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.

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.

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

Label the diagram:

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.

## FAQs - Frequently Asked Questions

### What are common trigonometric values?

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