# IB Mathematics (HL)/Trigonometry

## Topic 3: Core - Circular Functions and Trigonometry

Circular functions are based upon the unit circle.

### Length of an Arc

The arc length of an arc with radius r and angle θ radians between the two lines between the centre of the circle and the arc endpoints equals rθ.

This is merely a derivation from the circumference formula: The circumference of a circle equals 2πr, since 2π radians or 360 degrees is the angle of a full circle. l = θ r

### Identities

#### Pythagorean Identities

$\sin^2 x + \cos^2 x= 1$

$1 + \tan^2 x = \sec^2 x$

$1 + \cot^2 x = \csc^2 x$

### Sine and Cosine Rules and Triangle Areas

For any triangle with side lengths a, b and c, and internal angles A, B and C at the opposing corners, the following identites hold.

#### Sine Rule

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

or equivalently

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

#### Cosine Rule

$a^2=b^2+c^2-2bc\cos A$

$b^2=a^2+c^2-2ac\cos B$

$c^2=a^2+b^2-2ab\cos C$

#### Area of a Triangle

The area of the triangle equals

$\frac{1}{2}ab\sin C= \frac{1}{2}ac\sin B=\frac{1}{2}bc\sin A.$