01.5-Complementary

Complementary Angles

Recall that two angles are complementary if they add to $90^\circ \big( \text{ or } \dfrac{\pi}{2} \big)$

For example, $60^\circ \text{ and } 30^\circ$ are complementary because $60^\circ + 30^\circ = 90^\circ$

In radians: $\dfrac{\pi}{3} + \dfrac{\pi}{6} = \dfrac{\pi}{2}$

Complementary Angles in Trigonometry

The sine of an angle is equal to the cosine of its complement.

• $\sin(\theta) = \cos(90^\circ - \theta)$
• $\sin(\theta) = \cos(\dfrac{\pi}{2} - \theta)$

For example:

• In degrees, $\sin(80^\circ) = \cos(10^\circ)$
• In radians: $\sin \Big(\dfrac{\pi}{8} \Big) = \cos \Big( \dfrac{3\pi}{8} \Big)$

Cotangent

The cotangent (cot) of an angle is defined as the reciprocal of the tangent.

Or: $\cot(\theta) = \dfrac{1}{\tan(\theta)}$

The tangent of an angle is equal to the cotangent of its complement

• $\tan(\theta) = \cot(90^\circ - \theta) = \dfrac{1}{\tan(90^\circ - \theta)}$
• $\tan(\theta) = \cot \Big( \dfrac{\pi}{2}-\theta \Big)$

For example:

• In degrees, $\tan(18^\circ) = \cot(72^\circ)$
• In radians: $\tan \Big(\dfrac{\pi}{10} \Big) = \cot \Big( \dfrac{4\pi}{10} \Big)$

The same relationship holds in the other quadrants, but watch for the negatives.

In all of these examples,

• $(90^\circ - \theta) + (\theta) = 90^\circ \; \; \textit{so}$
• $\sin(90^\circ - \theta) = \cos(\theta)$

• $(90^\circ + \theta) + (-\theta) = 90^\circ \; \; \textit{so}$
• $\sin(90^\circ + \theta) = \cos(-\theta) = \cos(\theta)$

• $(270^\circ - \theta) + (-180^\circ + \theta) = 90^\circ \; \; \textit{so}$
• $\sin(270^\circ - \theta) = \cos(-180^\circ + \theta) = -\cos(\theta)$

• $(270^\circ + \theta) + (-180^\circ - \theta) = 90^\circ \; \; \textit{so}$
• $\sin(270^\circ + \theta) = \cos(-180^\circ - \theta) = -\cos(\theta)$
Notice that adding/subtracting the angles in each entry gives either $\dfrac{\pi}{2} \text{ or } \dfrac{3\pi}{2}$.