Complementary, Supplementary, and Opposite Angles

For all of these formulas, we will use the difference identity for cosine to prove them (and avoid circularity). However, their truth is easily seen via an appropriate diagram.


Complementary Angles

The complement of an angle $\theta$ is the angle that when combined $\theta$ produces a right angle ($90$ degree angle), i.e. it is $90^{\circ} - \theta$.

First, we prove the identity for cosine.

\[\begin{align} \cos(\pi/2 - \theta) &= \cos(\pi/2)\cos(\theta) + \sin(\pi/2)\sin(\theta) \\[10pt] &= 0 \cdot \cos(\theta) + 1 \cdot \sin(\theta) \\[10pt] &= \sin(\theta) \end{align}\]

Then, we use this result to prove the identity for sine.

\[\begin{align} \sin(\pi/2 - \theta) &= \cos(\pi/2 - (\pi/2 - \theta)) \\[10pt] &= \cos(\theta) \end{align}\]

Now, that we’ve shown the result for sine and cosine, all other functions are forced using the reciprocal and ratio identities. I just assert the results below.

\[\begin{align} &\sec(\pi/2 - \theta) = \csc(\theta) \qquad&&\csc(\pi/2 - \theta) = \sec(\theta) \\[10pt] &\tan(\pi/2 - \theta) = \cot(\theta) \qquad&&\cot(\pi/2 - \theta) = \tan(\theta) \end{align}\]

These identities confirm the dual nature of \(\{ \sin \theta, \sec \theta, \tan \theta \}\) and \(\{\cos \theta, \csc \theta, \cot \theta \}\), as we saw in the post Intuition of Trigonometric Functions. First, on the unit circle, they are symmetric about the line $y = x$. Second, when graphing these they are really the same functions, just shifted by $90^{\circ}$.


Supplementary Angles

The supplement of an angle $\theta$ is the angle that when combined $\theta$ produces a straight angle ($180$ degree angle), i.e. it is $180^{\circ} - \theta$.

We prove this similarly to complementary angles.

\[\begin{align} \cos(\pi - \theta) &= \cos(\pi)\cos(\theta) + \sin(\pi)\sin(\theta) \\[10pt] &= -1 \cdot \cos(\theta) + 0 \cdot \sin(\theta) \\[10pt] &= - \cos(\theta) \end{align}\]

Now, we just use some clever algebra and the complementary identity for cosine.

\[\begin{align} \sin(\pi - \theta) &= \sin(\pi/2 - (\theta - \pi/2)) \\[10pt] &= \cos(\theta - \pi/2) \\[10pt] &= \cos(\theta)\cos(\pi/2) + \sin(\theta)\sin(\pi/2) \\[10pt] &= \cos(\theta) \cdot 0 + \sin(\theta) \cdot 1 \\[10pt] &= \sin(\theta) \end{align}\]

Again, since we’ve shown the result for sine and cosine, all other functions are forced using the reciprocal and ratio identities. I just assert the results below.

\[\begin{align} &\sec(\pi - \theta) = -\sec(\theta) \qquad&&\csc(\pi - \theta) = \csc(\theta) \\[10pt] &\tan(\pi - \theta) = -\tan(\theta) \qquad&&\cot(\pi - \theta) = -\cot(\theta) \end{align}\]



Opposite Angles

Recall that $\theta \equiv \theta + 2 \pi k$ for any integer $k$, and in particular $(2 \pi - \theta) \equiv - \theta$.

Using geometry and the symmetry from the figure above, the result is obvious, but let’s prove it rigorously.

\[\begin{align} \cos(2\pi -\theta) = \cos(-\theta) &= \cos(0 - \theta) \\[10pt] &= \cos(0)\cos(\theta) + \sin(0)\sin(\theta) \\[10pt] &= 1 \cdot \cos(\theta) + 0 \cdot \sin(\theta) \\[10pt] &= \cos(\theta) \end{align}\]

Here, we use the fact that $-\pi/2 \equiv (-\pi/2 + 2 \pi) = 3\pi/2$.

\[\begin{align} \sin(2\pi -\theta) = \sin(-\theta) &= \cos(\pi/2 - (-\theta)) \\[10pt] &= \cos(\theta - (-\pi/2)) \\[10pt] &= \cos(\theta)\cos(-\pi/2) + \sin(\theta)\sin(-\pi/2) \\[10pt] &= \cos(\theta)\cos(3\pi/2) + \sin(\theta)\sin(3\pi/2) \\[10pt] &= \cos(\theta) \cdot 0 + \sin(\theta) \cdot (-1) \\[10pt] &= - \sin(\theta) \end{align}\]

Again, we can now assert the results for all the other functions

\[\begin{align} &\sec(2\pi -\theta) = \sec(- \theta) = \sec(\theta) \qquad&&\csc(2\pi -\theta) = \csc(- \theta) = -\csc(\theta) \\[10pt] &\tan(2\pi -\theta) = \tan(- \theta) = -\tan(\theta) \qquad&&\cot(2\pi -\theta) = \cot(- \theta) = -\cot(\theta) \end{align}\]


These identities seem trivial, but they actually have profound results. They show that $\cos(\cdot)$ and $\sec(\cdot)$ are even functions while $\sin(\cdot)$, $\csc(\cdot)$, $\tan(\cdot)$, and $\cot(\cdot)$ are odd functions. These facts are used a lot in higher mathematics, especially when taking integrals.

Trigonometry Series