Trigonometric Functions of Supplementary Angles

Supplementary angles play an important role in trigonometry because they allow many trigonometric expressions to be simplified quickly. If two angles are supplementary, one can be written as π - α. In this case, the following identities apply: $$ \sin(\pi-\alpha) = \sin(\alpha) $$ $$ \cos(\pi-\alpha) = -\cos(\alpha) $$ $$ \tan(\pi-\alpha) = -\tan(\alpha) $$ $$ \cot(\pi-\alpha) = -\cot(\alpha) $$

These identities show that supplementary angles share the same sine value, while cosine, tangent, and cotangent keep the same magnitude but change sign.

Why Do These Identities Work?

To understand these relationships, consider an angle α and its supplementary angle π - α on the unit circle.

The two angles are supplementary because together they form a straight angle:

$$ \alpha + (\pi - \alpha) = \pi $$

On the unit circle, these angles determine two right triangles that are congruent. They have the same hypotenuse, the same acute angle α, and a right angle.

As a result, the corresponding sides of the triangles have the same lengths.

The Sine of Supplementary Angles

The sine of an angle corresponds to its vertical coordinate on the unit circle. Since the two triangles have equal vertical sides, the sine values are identical.

$$ \sin \alpha = \sin(\pi - \alpha) $$

In other words, supplementary angles always have the same sine.

The Cosine of Supplementary Angles

The cosine corresponds to the horizontal coordinate on the unit circle. The horizontal distances have the same magnitude, but they lie on opposite sides of the y-axis.

For this reason, the cosine values have opposite signs:

$$ \cos(\pi - \alpha) = -\cos \alpha $$

Therefore, supplementary angles have cosine values with equal magnitudes but opposite signs.

The Tangent of Supplementary Angles

The tangent is defined as the ratio of sine to cosine:

$$ \tan(\pi - \alpha) = \frac{\sin(\pi - \alpha)}{\cos(\pi - \alpha)} $$

Substituting the identities for sine and cosine gives:

$$ \tan(\pi - \alpha) = \frac{\sin \alpha}{-\cos \alpha} = -\tan \alpha $$

So, the tangent of a supplementary angle is the negative of the original tangent.

The Cotangent of Supplementary Angles

The cotangent is defined as the ratio of cosine to sine:

$$ \cot(\pi - \alpha) = \frac{\cos(\pi - \alpha)}{\sin(\pi - \alpha)} $$

Using the same identities, we obtain:

$$ \cot(\pi - \alpha) = \frac{-\cos \alpha}{\sin \alpha} = -\cot \alpha $$

Thus, the cotangent of a supplementary angle is also the negative of the original cotangent.

Summary of the Identities

For supplementary angles α and π - α:

$$ \sin(\pi-\alpha)=\sin(\alpha) $$

$$ \cos(\pi-\alpha)=-\cos(\alpha) $$

$$ \tan(\pi-\alpha)=-\tan(\alpha) $$

$$ \cot(\pi-\alpha)=-\cot(\alpha) $$

These formulas are especially useful when evaluating trigonometric functions of angles in the second quadrant, where supplementary angles frequently appear.

Example: Calculating sin(150°)

Let us use these identities to calculate:

$$ \sin 150^\circ $$

Since 150° can be written as 180° - 30°, we have:

$$ \sin 150^\circ = \sin(180^\circ - 30^\circ) $$

In radians:

$$ \sin 150^\circ = \sin\left(\pi-\frac{\pi}{6}\right) $$

Applying the identity for supplementary angles:

$$ \sin\left(\pi-\frac{\pi}{6}\right)=\sin\left(\frac{\pi}{6}\right) $$

Therefore:

$$ \sin 150^\circ = \sin 30^\circ $$

Since:

$$ \sin 30^\circ = \frac{1}{2} $$

it follows that:

$$ \sin 150^\circ = \frac{1}{2} $$

This simple example illustrates how identities involving supplementary angles can make trigonometric calculations much easier.