# 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:

#### 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.