Trigonometry is used in many fields, such as construction, surveying, engineering, navigating and many other areas in research as experimental physics.

Trigonometry involves **right-angled triangle** extensively and there are ratios between the three sides that are useful in many ways. In order to refer to these ratios, the sides are named in relation to the angle being studied:

- the
**hypotenuse**is the longest side, and is always opposite the right angle - the side opposite to the studied angle marked '' (theta) is the
**opposite** - the
**adjacent**side is next to the angle marked ''

The trigonometric ratios are:

You may have seen these ratios in your scientific calculators. To remember these ratios easily, use the abbreviations: **SOH CAH TOA**.

**SOH **= **S**ine **O**pposite Hypotenuse

**CAH **= **C**osine **A**djacent **H**ypotenuse

**TOA **= **T**angent **O**pposite **A**djacent

As well as the basic ratios, there are **invese** trigonometric ratios:

It is also useful to understand the complementary angles in a right-angled triangle:

In

### Examples:

1. Find the exact values for cos* **x*, tan *x* and cosec *x.*

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2. If find .

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If then above represents the triangle. The length of hypotenuse DF could be foudnm by *Pythagoras' Theorem*.

Hence,

3. Simplify

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*Think:** Why does tan 90° result in ?*

## Advanced Trigonometric Ratios

1. Find the exact value of .

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2. Find the exact value of .

(Refer to Unit Circle)

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The angle is in the 2nd quadrant, and tan is negative in the 2nd quadrant.