Trigonometric Identities for the Associated Angles α and 3π/2 - α

In trigonometry, the associated angles α and 3π/2 - α are linked by a set of useful identities that make it easier to evaluate trigonometric functions by reducing them to equivalent expressions involving a simpler angle: $$ \sin\left(\frac{3\pi}{2}-\alpha\right)=-\cos\alpha $$ $$ \cos\left(\frac{3\pi}{2}-\alpha\right)=-\sin\alpha $$ $$ \tan\left(\frac{3\pi}{2}-\alpha\right)=\cot\alpha $$ $$ \cot\left(\frac{3\pi}{2}-\alpha\right)=\tan\alpha $$

These identities belong to the broader family of associated-angle formulas. They allow trigonometric functions of an angle in the third or fourth quadrant to be rewritten in terms of an acute reference angle, making calculations much more straightforward.

Why These Identities Work

To understand where these formulas come from, consider the angles α and 3π/2 - α on the unit circle.

Next, construct the two right triangles OAB and OCD inside the unit circle.

The triangles OAB and OCD are congruent because they share the same hypotenuse and have the same acute angle α.

As a consequence, their corresponding sides have the same length.

Note: The angles of a triangle add up to π radians (180°). In triangle OCD, one angle measures π/2 and another measures π/2 - α. Therefore, the third angle must be α: $$ \pi=\frac{\pi}{2}+\left(\frac{\pi}{2}-\alpha\right)+\alpha $$

Since the triangles are congruent, segment CD is equal in length to segment OB.

On the unit circle, CD represents the absolute value of the sine of 3π/2 - α, while OB represents the cosine of α.

Because the point lies below the x-axis, the sine value is negative. Therefore:

$$ -\sin\left(\frac{3\pi}{2}-\alpha\right)=\cos\alpha $$

Multiplying both sides by -1 gives:

$$ \sin\left(\frac{3\pi}{2}-\alpha\right)=-\cos\alpha $$

This proves the first identity.

Similarly, segment AB is equal in length to segment OD.

Here, OD represents the absolute value of the cosine of 3π/2 - α, while AB represents the sine of α.

Since the point lies to the left of the y-axis, the cosine value is negative:

$$ -\cos\left(\frac{3\pi}{2}-\alpha\right)=\sin\alpha $$

Multiplying both sides by -1 yields:

$$ \cos\left(\frac{3\pi}{2}-\alpha\right)=-\sin\alpha $$

This proves the second identity.

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 the ratio of sine to cosine:

$$ \tan\left(\frac{3\pi}{2}-\alpha\right) = \frac{\sin\left(\frac{3\pi}{2}-\alpha\right)} {\cos\left(\frac{3\pi}{2}-\alpha\right)} $$

Substituting the identities already derived:

$$ \tan\left(\frac{3\pi}{2}-\alpha\right) = \frac{-\cos\alpha}{-\sin\alpha} = \cot\alpha $$

Therefore:

$$ \tan\left(\frac{3\pi}{2}-\alpha\right)=\cot\alpha $$

The cotangent of an angle is the ratio of cosine to sine:

$$ \cot\left(\frac{3\pi}{2}-\alpha\right) = \frac{\cos\left(\frac{3\pi}{2}-\alpha\right)} {\sin\left(\frac{3\pi}{2}-\alpha\right)} $$

Substituting the same identities:

$$ \cot\left(\frac{3\pi}{2}-\alpha\right) = \frac{-\sin\alpha}{-\cos\alpha} = \tan\alpha $$

Hence:

$$ \cot\left(\frac{3\pi}{2}-\alpha\right)=\tan\alpha $$

Example: Calculating sin(210°)

Let us use these identities to evaluate the sine of 210°.

$$ \sin 210^\circ $$

First, rewrite 210° as:

$$ 210^\circ = 270^\circ - 60^\circ $$

Therefore:

$$ \sin 210^\circ = \sin(270^\circ-60^\circ) $$

In radians, this becomes:

$$ \sin\left(\frac{3\pi}{2}-\frac{\pi}{3}\right) $$

Since α = π/3, apply the identity:

$$ \sin\left(\frac{3\pi}{2}-\alpha\right) = -\cos\alpha $$

Substituting α = π/3 gives:

$$ \sin\left(\frac{3\pi}{2}-\frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) $$

Because:

$$ \cos\left(\frac{\pi}{3}\right)=\frac{1}{2} $$

we obtain:

$$ \sin\left(\frac{3\pi}{2}-\frac{\pi}{3}\right) = -\frac{1}{2} $$

Therefore:

$$ \sin 210^\circ=-\frac{1}{2} $$

Instead of evaluating the sine of a third-quadrant angle directly, we transformed it into the cosine of a first-quadrant angle, which is much easier to compute.

This is one of the main advantages of associated-angle identities: they simplify trigonometric calculations by reducing them to familiar reference angles.