Cofunction Identities from Complementary Angles

Introduction

Some trigonometric expressions involve angles whose measures add to a right angle. Such angles are called complementary. Their trigonometric values are closely related: the sine of one angle equals the cosine of its complement, and tangent and cotangent are paired in the same way.

These relationships are called cofunction identities. They follow naturally from the unit circle: exchanging the two coordinates of a point corresponds to reflecting it across the line \(y=x\), which changes its angle to its complement. We will derive the identities from this geometric picture, state the conditions under which the quotient identities are defined, and use them to evaluate exact values.

Complementary Angles

Two angles are complementary when their measures add to a right angle. In radians, a right angle has measure \(\frac{\pi}{2}\); in degrees, it has measure \(90^\circ\). Thus the complement of an angle \(\theta\) is \(\frac{\pi}{2}-\theta\) in radians, or \(90^\circ-\theta\) in degrees.

The identities below are not restricted to acute angles. The same relationships hold for any real \(\theta\), with the usual care that tangent and cotangent are defined only when their denominators are nonzero. When an angle is outside the first quadrant, its sine or cosine may be negative; the cofunction identities still retain the same signs because they express equality for the actual angles, not just equality of lengths.

Why Sine and Cosine Exchange

On the unit circle, the point corresponding to an angle \(\theta\) has coordinates \((\cos\theta,\sin\theta)\). Reflecting this point across the line \(y=x\) exchanges its coordinates, giving \((\sin\theta,\cos\theta)\).

The line \(y=x\) makes an angle of \(\frac{\pi}{4}\) with the positive horizontal axis. Reflection across a line at angle \(\frac{\pi}{4}\) sends a direction at angle \(\theta\) to a direction at angle \(2\left(\frac{\pi}{4}\right)-\theta=\frac{\pi}{2}-\theta\). Therefore the reflected point is also the unit-circle point for the complementary angle \(\frac{\pi}{2}-\theta\). Since the coordinates of a unit-circle point at angle \(\frac{\pi}{2}-\theta\) are \(\left(\cos\left(\frac{\pi}{2}-\theta\right),\sin\left(\frac{\pi}{2}-\theta\right)\right)\), matching coordinates gives the two identities.

This argument is a complete geometric derivation: reflection exchanges the coordinates, and its effect on the angle is to replace \(\theta\) with \(\frac{\pi}{2}-\theta\). The first coordinate equality says that the cosine of the complement equals the original sine; the second says that the sine of the complement equals the original cosine.

Deriving the Tangent and Cotangent Identities

Tangent and cotangent are quotients of the unit-circle coordinates. Where their denominators are nonzero, \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). The sine-cosine identities therefore produce the tangent-cotangent identities.

This calculation is valid when \(\sin\theta\ne0\). That is precisely the condition that both sides are defined: \(\cot\theta\) requires a nonzero sine, and the denominator of the complementary tangent is \(\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta\).

Likewise, when \(\cos\theta\ne0\),

Here \(\cos\theta\ne0\) is required on both sides: it is the denominator in \(\tan\theta\), and it equals \(\sin\left(\frac{\pi}{2}-\theta\right)\), the denominator in the complementary cotangent.

The complete set of cofunction identities and their domain requirements is summarized below.

Worked Exact Evaluations

A Common Mistake

A complement changes the angle, and the corresponding function changes too. It is incorrect to assume that changing an angle to its complement leaves the same trigonometric function unchanged. For example, the following calculation is wrong:

The angle on the left is \(\frac{\pi}{3}\), not \(\frac{\pi}{6}\). The sine-cosine identity gives the correct calculation:

For a direct check, \(\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\), not \(\frac{1}{2}\). A useful reminder is that complementary angles exchange sine and cosine, and exchange tangent and cotangent; they do not generally preserve the function name.

Also check domains before using a tangent-cotangent identity. For \(\theta=0\), the expression \(\tan\left(\frac{\pi}{2}-\theta\right)\) is \(\tan\left(\frac{\pi}{2}\right)\), which is undefined, and \(\cot\theta=\cot 0\) is also undefined. The identity relates the two functions where they exist; it does not make either one defined at excluded angles.

Why Cofunction Identities Are Useful

Cofunction identities allow an expression involving one trigonometric function to be rewritten using its paired function. This is especially useful when an angle is naturally written as a complement, or when a familiar exact value is available for the complementary angle. For instance, evaluating \(\cos\left(\frac{\pi}{3}\right)\) can be reduced to the standard value of \(\sin\left(\frac{\pi}{6}\right)\).

They also provide a way to check whether an expression has been interpreted correctly. If the angles add to \(\frac{\pi}{2}\), sine and cosine should exchange; tangent and cotangent should exchange when defined. If a calculation keeps the same function without another justification, it is worth checking the angle and the identity again.