# Trigonometry

Trigonometry is a branch of mathematics that concerns the relationships between the sides and angles of triangles. There are three main trigonometric functions: sine, cosine, and tangent.

## Right triangle

Right triangles are widely used in trigonometry. A right triangle is a triangle in which one angle has a measurement of 90° (a right angle), such as the triangle shown below.

The other angles of a right triangle are often represented by Greek letters, such as θ, α, and β. The sides of a right triangle are referenced as follows:

- Adjacent: the side next to the angle
- Opposite: the side opposite the angle
- Hypotenuse: the side opposite the right angle. It is also the longest side.

## Trigonometric functions

The main trigonometric functions are sine, cosine, and tangent. Sine, cosine, and tangent are often abbreviated as sin, cos, and tan. Trigonometric functions are also called circular functions.

Example:

What is the value of sin(45°), cos(45°), and tan(45°)?

Example:

Bob walked 300 m straight up on a 30° hill, how high did Bob climb?

Height = | distance walked × sin(30°) |

= | 300 × 0.5 |

= | 150 m |

Thus, Bob reached a height of 150 m after walking 300 m up the slope of the hill.

Secant, cotangent, and cosecant are also trigonometric functions, but they are rarely used.

## Inverse trigonometric functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions:

Given y = sine(θ), θ = arcsin(y) |

Given y = cosine(θ), θ = arccos(y) |

Given y = tangent(θ), θ = arctan(y) |

Arcsin, arccos, and arctan are the inverse functions of sine, cosine, and tangent respectively.

Example:

Given a right triangle with opposite length 3 and hypotenuse length 6, what is the value of θ?

## Non-right triangles

Unlike right triangles, the sine, cosine, and tangent of a non-right triangle cannot be used to calculate the length of the sides and angles of the triangle directly. Instead the following rules can be used.

### The sine rule

For an arbitrary triangle,

The sine rule can be used when one side and any two angles are known.

Example:

Given, a=3, B=70°, C=45°, find A, b, and c,

A = 180° - B - C = 180° - 70° - 45° = 65° |

### The cosine rule

For an arbitrary triangle,

a^{2} = b^{2} + c^{2} – 2bc·cos(A) |

b^{2} = a^{2} + c^{2} – 2ac·cos(B) |

c^{2} = a^{2} + b^{2} – 2ab·cos(C) |

The cosine rule can be used when one angle and the two sides next to it are known.

Example:

Given, A=45°, b=3, c=5, find a,

a^{2} | = b^{2} + c^{2} – 2bc·cos(A) |

= 3^{2} + 5^{2} + 2×3×5×cos(45°) | |

= 55.213 | |