Trigonometric Identities for Angles Differing by π/2

When two angles differ by π/2 radians (90°), their trigonometric functions are closely related. These relationships, known as trigonometric identities for angles differing by π/2, allow you to rewrite the sine, cosine, tangent, and cotangent of one angle in terms of the corresponding functions of the other. The identities are: $$ \sin\left(\frac{\pi}{2}+\alpha\right)=\cos(\alpha) $$ $$ \cos\left(\frac{\pi}{2}+\alpha\right)=-\sin(\alpha) $$ $$ \tan\left(\frac{\pi}{2}+\alpha\right)=-\cot(\alpha) $$ $$ \cot\left(\frac{\pi}{2}+\alpha\right)=-\tan(\alpha) $$

These formulas are particularly useful for simplifying trigonometric expressions and evaluating functions at special angles.

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.

The second angle is obtained by rotating the first angle by an additional 90°.

This rotation creates two congruent right triangles, OAB and OCD. They have the same hypotenuse and share the same acute angle α.

Because the triangles are congruent, their corresponding sides have the same length. In particular, segment OB is equal to segment OD.

On the unit circle, OB represents the cosine of α, while OD represents the sine of π/2 + α.

This immediately gives the first identity:

$$ \sin\left(\frac{\pi}{2}+\alpha\right)=\cos\alpha $$

A similar argument applies to segments AB and CD, which are also equal in length.

Since AB represents sin α and CD corresponds to the opposite of cos(π/2 + α), we obtain:

$$ \cos\left(\frac{\pi}{2}+\alpha\right)=-\sin\alpha $$

Together, these two identities explain how a 90° rotation exchanges the roles of sine and cosine while introducing a sign change where required.

Finding the Tangent and Cotangent Identities

Once the sine and cosine identities are known, the formulas for tangent and cotangent follow directly from their definitions.

The tangent of an angle is defined as:

$$ \tan\theta=\frac{\sin\theta}{\cos\theta} $$

Therefore,

$$ \tan\left(\frac{\pi}{2}+\alpha\right)=\frac{\sin\left(\frac{\pi}{2}+\alpha\right)}{\cos\left(\frac{\pi}{2}+\alpha\right)} $$

Substituting the previous identities gives:

$$ \tan\left(\frac{\pi}{2}+\alpha\right)=\frac{\cos\alpha}{-\sin\alpha}=-\cot\alpha $$

Hence:

$$ \tan\left(\frac{\pi}{2}+\alpha\right)=-\cot\alpha $$

The cotangent is defined as:

$$ \cot\theta=\frac{\cos\theta}{\sin\theta} $$

Applying the same substitutions:

$$ \cot\left(\frac{\pi}{2}+\alpha\right)=\frac{-\sin\alpha}{\cos\alpha}=-\tan\alpha $$

Therefore:

$$ \cot\left(\frac{\pi}{2}+\alpha\right)=-\tan\alpha $$

Example: Calculating sin 150°

Let us use these identities to calculate the sine of 150°.

First, rewrite the angle as:

$$ 150^\circ = 90^\circ + 60^\circ $$

Therefore,

$$ \sin 150^\circ=\sin(90^\circ+60^\circ) $$

In radians, this becomes:

$$ \sin 150^\circ=\sin\left(\frac{\pi}{2}+\frac{\pi}{3}\right) $$

Using the identity

$$ \sin\left(\frac{\pi}{2}+\alpha\right)=\cos\alpha $$

with α = π/3, we obtain:

$$ \sin 150^\circ=\cos\left(\frac{\pi}{3}\right) $$

Since

$$ \cos 60^\circ=\frac{1}{2} $$

it follows that

$$ \sin 150^\circ=\frac{1}{2} $$

So, the value of sin 150° is:

$$ \boxed{\sin 150^\circ=\frac{1}{2}} $$