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# Cofunction Identities — Formula, Proof, Examples

[#Trigonometry](/content/tag/trigonometry/index.html)

TL;DR  
The cofunction identities state that any trig function of θ \(θ\) equals the corresponding co-function of the complementary angle \(\frac{\pi}{2}-θ\) — six pairings that turn \(\sin(60°)\) into \(\cos(30°)\) without computation. This article gives the six identities, the right-triangle and unit-circle proof, three worked examples in degrees and radians, the application to simplifying expressions, and the common mistakes around the "co" prefix.

## Two Angles That Always Add to a Right Angle — and Six Identities That Follow

Every "co" in trigonometry (cosine, cotangent, cosecant) is short for "complementary" — the function of the complementary angle.

Two angles are **complementary** when they sum to \(\frac{\pi}{2}\) radians (90°). The **cofunction identities** say that for any angle \(θ\):

- \(\sin(\frac{\pi}{2}-θ)=\cos(θ)\) — and vice versa.
- \(\tan(\frac{\pi}{2}-θ)=\cot(θ)\) — and vice versa.
- \(\sec(\frac{\pi}{2}-θ)=\csc(θ)\) — and vice versa.

These six pairings are the algebraic statement of the right-triangle fact that _the two acute angles of any right triangle are complementary_.

## The Six Formulas

\[\begin{align*}  
\sin\left(\frac{\pi}{2}-θ\right) &= \cos(θ)  \\
\cos\left(\frac{\pi}{2}-θ\right) &= \sin(θ)  \\
\tan\left(\frac{\pi}{2}-θ\right) &= \cot(θ)  \\
\cot\left(\frac{\pi}{2}-θ\right) &= \tan(θ)  \\
\sec\left(\frac{\pi}{2}-θ\right) &= \csc(θ)  \\
\csc\left(\frac{\pi}{2}-θ\right) &= \sec(θ)  
\end{align*}\]

In degrees the same six identities hold with \(\frac{\pi}{2}\) replaced by 90°.

### **Quick Facts:**
- **Three cofunction pairs:** \((\sin, \cos), (\tan, \cot), (\sec, \csc)\).
- **Domain:** the sine/cosine identities hold for all real \(θ\); tangent/cotangent and secant/cosecant identities hold wherever both sides are defined.
- **Reflection across \(θ=\frac{\pi}{4}\):** the cofunction identities are the algebraic statement of reflecting the graph of \(\sin\) across \(x=\frac{\pi}{4}\) to get the graph of \(\cos\).

## Double-Anchoring — Right Triangle and Unit Circle

The cleanest proof of the cofunction identities sits in the right triangle and is mirrored on the unit circle.

**From the right triangle.** In a right triangle with one acute angle \(θ\), the other acute angle is \(\frac{\pi}{2}-θ\) (since the angles sum to \(\pi\) and one is \(\frac{\pi}{2}\)). For the angle \(θ\): opposite leg = a, adjacent leg = b, hypotenuse = c. Thus, \(\sin(θ)=\frac{a}{c}\) and \(\cos(θ)=\frac{b}{c}\). For the angle \(\frac{\pi}{2}-θ\): the opposite leg is now b, the adjacent leg is now a. Thus, \(\sin(\frac{\pi}{2}-θ)=\frac{b}{c}=\cos(θ)\) and \(\cos(\frac{\pi}{2}-θ)=\frac{a}{c}=\sin(θ)\).

**From the unit circle.** A point at angle \(θ\) has coordinates \((\cos(θ), \sin(θ))\). A point at angle \(\frac{\pi}{2}-θ\) has coordinates \((\cos(\frac{\pi}{2}-θ), \sin(\frac{\pi}{2}-θ))\). Reflecting \((\cos(θ), \sin(θ))\) swaps its coordinates to \((\sin(θ), \cos(θ))\). Thus, the identities hold.

## Three Worked Examples of Cofunction Identities

**Quick.** Express \(\sin(60°)\) in terms of cosine of a complementary angle.

The complement of 60° is 30°. Applying the cofunction identity gives:  \(\sin(60°)=\cos(30°)\).

**Final answer:** \(\sin(60°)=\cos(30°)=\frac{\sqrt{3}}{2}\).

**Standard (Wrong Path First — Where Students Lose the Mark).** Evaluate \(\sin(235°)+\sin(255°)\) without a calculator.
- _The wrong path._ A student tries to use the Pythagorean identity directly, incorrectly.
- _The correct path._ Instead, note that both angles are complementary:
  \(\sin(255°)=\sin(90° - 35°)=\cos(35°)\).

Final result: \(\sin(235°)+\sin(255°)=1\).

**Stretch.** Simplify \(\frac{\tan(\frac{\pi}{2}-θ) \cdot \sec(θ)}{\csc(θ)}\) in terms of basic trig functions.
- Applying the cofunction identity yields: \(\frac{\cot(θ) \cdot \sec(θ)}{\csc(θ)}\).
- The final answer is 1.

## Cofunction Identities: Where Cofunction Identities Quietly Power Real Work

These identities show up wherever a sine-related calculation needs to be re-expressed in cosine form or vice versa:
- **Phasor analysis.** AC circuits computing reactive power.
- **Signal processing.** Digital communication systems.
- **Computer graphics.** 2D rotation matrices. 
- **Surveying.** Converting between azimuths.
- **Optics.** Light polarization components.

## The Mathematicians Behind the "Co" in Cosine

The word _cosine_ literally means _"sine of the complement"_.

**Edmund Gunter** coined the term _co.sinus_ in 1620. He paired each angle's sine with the sine of its complement.
**Aryabhata** tabulated complementary sines six centuries earlier. Their work ties the concept of sine to its complement closely.

## Three Errors That Cost the Most Marks In Cofunction Identities

1. **Confusing "co-" with "negative."**
2. **Mismatching angle measures inside the identity.**
3. **Restricting the identity to acute angles only.**
4. **Applying the cofunction identity incorrectly for non-complementary angles.**

## The Short Version

Cofunction identities pair each trig function with its complementary partner:
- \(\sin \leftrightarrow \cos\)
- \(\tan \leftrightarrow \cot\)
- \(\sec \leftrightarrow \csc\)  
Cofunction identities simplify trigonometric expressions involving complements and are foundational results in trigonometry.