Trigonometry: Co-Function Identities

The cofunction identities state that $$\begin{align*} \sin (90^{\circ}-x) &= \cos x\\ \cos (90^{\circ}-x) &= \sin x\\ \tan (90^{\circ}-x) &= \cot x\\ \cot (90^ {\circ}-x) &= \tan x\\ \sec (90^{\circ}-x) &= \csc x\\ \csc (90^{\circ}-x) &= \sec x\\ \end{align*}$$

The first two identities follow from the unit circle, the others following from the first two via the fact that $\tan \theta = \frac{ \sin \theta}{\cos \theta}$, $\cot = \frac{ \cos \theta}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\csc = \frac{1}{\sin \theta}$.

These relations are very useful in simplifying calculations in both geometry and complex numbers, proving other identities, and evaluating trigonometric functions, though the first 2 are really the most commonly used ones.

Cofunction Identities

There are six negative angle identities you need to memorize!

• Negative angle identities
• Co-function identities (this lesson)
• Angle and sum differences identities
• Double angle identities
• Half angle identities

You'll recognize sine and cosine, cosecant and secant, and tangent and cotangent.

Memorizing these cofunction identities will help you to solve trigonometric equations in the future.

(Video) Co-Function Identities

by Mario's Math Tutoring

In this video, Mario's Math Tutoring will help you understand the cofunction identities.

Summary

We can think of the cofunction identities in pairs. First,

$$sin (\frac{\pi}{2} - \theta) = cos(\theta)$$ $$cos (\frac{\pi}{2} - \theta) = sin(\theta)$$

We can also pair together:

$$sec (\frac{\pi}{2} - \theta) = csc(\theta)$$ $$csc (\frac{\pi}{2} - \theta) = sec(\theta)$$

Lastly, we can pair together:

$$tan (\frac{\pi}{2} - \theta) = cot (\theta)$$ $$cot (\frac{\pi}{2} - \theta) = tan (\theta)$$

Let's think about why this works. Think of the fact that all angles in a right triangle must equal $180^{\circ}$. If the right angle already comprises of the $90^{\circ}$, then the rest of the $90^{\circ}$ must be distributed in the other two angles. (Imagine acute, obtuse, and right angles!)

Let's walk through an example.

Example #1

$$\frac{\text{tan} (\frac{\pi}{2} - \theta)}{\text{cos} (\theta)}$$

At first, this expression looks hard and complicated. However, we know that $tan (\frac{\pi}{2} - \theta) = cot (\theta)$, so our numerator becomes $cot (\theta)$. Our new fraction is:

$$\frac{cot(\theta)}{cos (\theta)}$$

We also know that $cot (\theta) = \frac{cos (\theta)}{sin (\theta)}$. If we simplify our new fraction, we will find that the answer is $csc (\theta)$.

Try the next example on your own, then watch the video to see how you did!

Example #2

$$sin^2 (10^{\circ}) + sin^2(80^{\circ})$$