Negative Angles and Angles ≥ 360°
Introduction
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.
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°.
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°.
Explore!
Explore!
Use the applet below to explore the relationship between Negative Angles and Angles greater than 360° with angles between 0° and 360°.
Use the applet below to explore the relationship between Negative Angles and Angles greater than 360° with angles between 0° and 360°.
Trigonometric Ratios for Negative Angles
Trigonometric Ratios for Negative Angles
Negative Angle formulas are similar to the 4th Quadrant formulas.
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:
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
sin(– A) = – sin A
cos(– A) = + cos A
cos(– A) = + cos A
tan(– A) = – tan A
tan(– A) = – tan A
Explore!
Explore!
Use the applet below to visualise how rotational symmetry of the sine and tangent graph about the origin, as well as how the cosine graph is a reflection across the y-axis results in the 4th Quadrant formulas.
Use the applet below to visualise how rotational symmetry of the sine and tangent graph about the origin, as well as how the cosine graph is a reflection across the y-axis results in the 4th Quadrant formulas.
Trigonometric Ratios for Angles ≥ 360°
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:
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
sin(A ± 360° × n) = sin A
cos(A ± 360° × n) = cos A
cos(A ± 360° × n) = cos A
tan(A ± 180° × n) = tan A
tan(A ± 180° × n) = tan A
where n is an integer.
where n is an integer.