Trigonometric Functions of Negative Angles
Negative angles have simple and useful relationships with their corresponding positive angles. In trigonometry, these relationships are described by the following identities: $$ \sin(-\alpha) = -\sin(\alpha) $$ $$ \cos(-\alpha) = \cos(\alpha) $$ $$ \tan(-\alpha) = -\tan(\alpha) $$ $$ \cot(-\alpha) = -\cot(\alpha) $$
These formulas make it easy to evaluate the trigonometric functions of a negative angle once the values for the corresponding positive angle are known.
Why do these identities work?
To understand these formulas, it helps to look at the unit circle.
Consider an angle \(\alpha\) and the corresponding negative angle \(-\alpha\).
On the unit circle, a positive angle is measured counterclockwise, while a negative angle is measured clockwise. The points associated with \(\alpha\) and \(-\alpha\) are mirror images of each other across the x-axis.
This symmetry is the key to understanding the trigonometric identities for negative angles.
Sine of a Negative Angle
The sine of an angle corresponds to the y-coordinate of the point on the unit circle.
Because the points associated with \(\alpha\) and \(-\alpha\) are reflected across the x-axis, their y-coordinates have the same magnitude but opposite signs.
Therefore:
$$ \sin(-\alpha) = -\sin(\alpha) $$
This property shows that sine is an odd function.
Cosine of a Negative Angle
The cosine of an angle corresponds to the x-coordinate of the point on the unit circle.
When a point is reflected across the x-axis, its x-coordinate does not change. As a result, the cosine values of \(\alpha\) and \(-\alpha\) are identical.
Thus:
$$ \cos(-\alpha) = \cos(\alpha) $$
This property shows that cosine is an even function.
Tangent of a Negative Angle
The tangent of an angle is defined as the ratio of sine to cosine:
$$ \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} $$
Replacing \(\alpha\) with \(-\alpha\) gives:
$$ \tan(-\alpha) = \frac{\sin(-\alpha)}{\cos(-\alpha)} $$
Using the identities for sine and cosine:
$$ \tan(-\alpha) = \frac{-\sin(\alpha)}{\cos(\alpha)} $$
Therefore:
$$ \tan(-\alpha) = -\tan(\alpha) $$
Like sine, tangent is also an odd function.
Cotangent of a Negative Angle
The cotangent of an angle is defined as the ratio of cosine to sine:
$$ \cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} $$
Replacing \(\alpha\) with \(-\alpha\) gives:
$$ \cot(-\alpha) = \frac{\cos(-\alpha)}{\sin(-\alpha)} $$
Substituting the identities for sine and cosine:
$$ \cot(-\alpha) = \frac{\cos(\alpha)}{-\sin(\alpha)} $$
Hence:
$$ \cot(-\alpha) = -\cot(\alpha) $$
Cotangent is therefore an odd function as well.
Example: Calculating sin(-30°)
Let's use the identity for the sine of a negative angle to compute:
$$ \sin(-30^\circ) $$
Since:
$$ \sin(-\alpha) = -\sin(\alpha) $$
we can write:
$$ \sin(-30^\circ) = -\sin(30^\circ) $$
The sine of \(30^\circ\) is:
$$ \sin(30^\circ) = \frac{1}{2} $$
Therefore:
$$ \sin(-30^\circ) = -\frac{1}{2} $$
This example illustrates a useful principle: once you know the trigonometric value of a positive angle, the corresponding value for the negative angle can often be found immediately using the identities above.
Summary
The trigonometric functions of negative angles follow a simple pattern:
$$ \sin(-\alpha) = -\sin(\alpha) $$
$$ \cos(-\alpha) = \cos(\alpha) $$
$$ \tan(-\alpha) = -\tan(\alpha) $$
$$ \cot(-\alpha) = -\cot(\alpha) $$
In other words, sine, tangent, and cotangent change sign when the angle becomes negative, while cosine remains unchanged. These identities are fundamental in trigonometry and are widely used in algebraic manipulations, graph analysis, and the solution of trigonometric equations.
