Negative Angles and Angles ≥ 360°

Introduction

What does a negative angle look like? Can an angle be bigger than 360°? In trigonometry, we defined positive angles as being measured in an anti-clockwise direction from the horizontal axis hence a negative angle is just measuring the angle in a clockwise direction.

For angles bigger than 360°, you can think of it as a rotation that is bigger than one complete turn. For example, the wheel of a bus turn round and round and each round is 360°.

Trigonometric Ratios for Negative Angles

Negative Angle formulas are similar to the 4th Quadrant formulas.

It is also a result of the parity of the trigonometric functions. Sine and Tangent are odd functions (Rotational Symmetry about origin) while cosine is an even function (Reflection across y-Axis). This can be summarised in the formulas below:

sin(– A) = – sin A

cos(– A) = + cos A

tan(– A) = – tan A

Trigonometric Ratios for Angles ≥ 360°

Because the period of sine and cosine is 360°, adding or subtracting 360° to an angle does not change its trigonometric ratio. For the case of tangent, the period is 180° so adding or subtracting 180° does not change the tangent ratio. This can be summarised in the formulas below:

sin(A ± 360° × n) = sin A

cos(A ± 360° × n) = cos A

tan(A ± 180° × n) = tan A

where n is an integer.