Co-Function Identities for Complementary Angles
In trigonometry, complementary angles α and π/2 - α are connected by a set of useful relationships known as co-function identities: $$ \sin\left(\frac{\pi}{2}-\alpha\right)=\cos(\alpha) $$ $$ \cos\left(\frac{\pi}{2}-\alpha\right)=\sin(\alpha) $$ $$ \tan\left(\frac{\pi}{2}-\alpha\right)=\cot(\alpha) $$ $$ \cot\left(\frac{\pi}{2}-\alpha\right)=\tan(\alpha) $$
These identities allow you to rewrite a trigonometric function of an angle in terms of the corresponding co-function of its complement. They are particularly useful when simplifying expressions, proving identities, and evaluating trigonometric functions without a calculator.
The angles α and π/2 - α are associated angles. More precisely, they are complementary angles because their sum is π/2 radians, or 90°.
$$ \alpha+\left(\frac{\pi}{2}-\alpha\right)=\frac{\pi}{2} $$
Why Do These Identities Work?
A simple way to understand these relationships is to look at the angles α and π/2 - α on the unit circle.
Inside the unit circle, we can construct two right triangles, OAB and OCD.
In each triangle, two angles are already known. Since the angles of a triangle always add up to π radians (180°), the third angle can be found immediately.
Note. In triangle OAB, one acute angle is α and another angle is a right angle. Therefore, the remaining angle must be π/2 - α. In triangle OCD, one acute angle is π/2 - α and another angle is a right angle, so the remaining angle must be α.
The two triangles have the same angles and the same hypotenuse. Since both hypotenuses are radii of the unit circle, the triangles are congruent.
As a result, corresponding sides have equal lengths.
For example, segment OB represents the cosine of α, while segment CD represents the sine of π/2 - α.
Because these segments have the same length,
$$ \sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha $$
Likewise, segment AB represents the sine of α, while segment OD represents the cosine of π/2 - α.
Since these segments are also equal,
$$ \cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha $$
Co-Function Identities for Tangent and Cotangent
Once the identities for sine and cosine are known, the corresponding formulas for tangent and cotangent follow directly from their definitions.
The tangent of an angle is the ratio of its sine to its cosine:
$$ \tan\left(\frac{\pi}{2}-\alpha\right)=\frac{\sin\left(\frac{\pi}{2}-\alpha\right)}{\cos\left(\frac{\pi}{2}-\alpha\right)} $$
Substituting the identities above gives:
$$ \tan\left(\frac{\pi}{2}-\alpha\right)=\frac{\cos\alpha}{\sin\alpha}=\cot\alpha $$
Therefore, the tangent of an angle is equal to the cotangent of its complement.
The cotangent of an angle is the ratio of its cosine to its sine:
$$ \cot\left(\frac{\pi}{2}-\alpha\right)=\frac{\cos\left(\frac{\pi}{2}-\alpha\right)}{\sin\left(\frac{\pi}{2}-\alpha\right)} $$
Substituting the same identities gives:
$$ \cot\left(\frac{\pi}{2}-\alpha\right)=\frac{\sin\alpha}{\cos\alpha}=\tan\alpha $$
Therefore, the cotangent of an angle is equal to the tangent of its complement.
Worked Example
Let's use a co-function identity to calculate the sine of 30°.
$$ \sin 30^\circ $$
Notice that 30° is the complement of 60°:
$$ 30^\circ = 90^\circ - 60^\circ $$
Therefore,
$$ \sin(30^\circ)=\sin(90^\circ-60^\circ) $$
In radians, this becomes:
$$ \sin\left(\frac{\pi}{2}-\frac{\pi}{3}\right) $$
Using the co-function identity
$$ \sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha $$
with α = π/3, we obtain:
$$ \sin\left(\frac{\pi}{2}-\frac{\pi}{3}\right)=\cos\left(\frac{\pi}{3}\right) $$
Since
$$ \cos\left(\frac{\pi}{3}\right)=\frac{1}{2} $$
it follows that
$$ \sin\left(\frac{\pi}{2}-\frac{\pi}{3}\right)=\frac{1}{2} $$
Therefore,
$$ \sin 30^\circ=\frac{1}{2} $$
This example illustrates one of the main advantages of co-function identities: they allow you to replace a trigonometric function with an equivalent one that may be easier to evaluate.
