Trigonometric Identities for α and α + 3π/2
In trigonometry, the angles α and α + 3π/2 are associated angles. Their trigonometric functions are linked by a set of simple identities that make it possible to rewrite complex expressions in terms of familiar first-quadrant angles: $$ \sin \left( \alpha + \frac{3\pi}{2} \right) = -\cos \alpha $$ $$ \cos \left( \alpha + \frac{3\pi}{2} \right) = \sin \alpha $$ $$ \tan \left( \alpha + \frac{3\pi}{2} \right) = -\cot \alpha $$ $$ \cot \left( \alpha + \frac{3\pi}{2} \right) = -\tan \alpha $$
These identities are especially useful when simplifying trigonometric expressions, solving equations, or evaluating functions without a calculator.
Why Are the Angles α and α + 3π/2 Related?
To understand these identities, consider the angles α and α + 3π/2 on the unit circle.
The angle α + 3π/2 is obtained by rotating the angle α by 270°, or 3π/2 radians.
This rotation creates two right triangles, OAB and OCD, that are congruent because they have the same hypotenuse length (OA = OC) and the same acute angle α.
Since congruent triangles have equal corresponding sides, segment OB has the same length as segment OD.
Segment OB represents the cosine of α on the positive x-axis. Segment OD represents the magnitude of the sine of α + 3π/2 on the negative y-axis.
Therefore:
$$ -\sin \left( \alpha + \frac{3\pi}{2} \right) = \cos \alpha $$
Multiplying both sides by -1 gives:
$$ \sin \left( \alpha + \frac{3\pi}{2} \right) = -\cos \alpha $$
In other words, the sine of α + 3π/2 is equal to the negative of the cosine of α.
A similar argument applies to segments AB and CD, which are also equal in length.
Segment AB represents the sine of α, while segment CD represents the cosine of α + 3π/2.
As a result:
$$ \cos \left( \alpha + \frac{3\pi}{2} \right) = \sin \alpha $$
This shows that a 270° rotation transforms sine into cosine and cosine into sine, while also changing the sign where required by the quadrant.
Deriving 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 is the ratio of sine to cosine:
$$ \tan \left( \alpha + \frac{3\pi}{2} \right) = \frac{\sin\left(\alpha + \frac{3\pi}{2}\right)} {\cos\left(\alpha + \frac{3\pi}{2}\right)} $$
Substituting the identities obtained above:
$$ \tan \left( \alpha + \frac{3\pi}{2} \right) = \frac{-\cos \alpha}{\sin \alpha} = -\cot \alpha $$
Therefore, the tangent of α + 3π/2 is the negative of the cotangent of α.
The cotangent is the ratio of cosine to sine:
$$ \cot \left( \alpha + \frac{3\pi}{2} \right) = \frac{\cos\left(\alpha + \frac{3\pi}{2}\right)} {\sin\left(\alpha + \frac{3\pi}{2}\right)} $$
Replacing sine and cosine with their equivalent expressions:
$$ \cot \left( \alpha + \frac{3\pi}{2} \right) = \frac{\sin \alpha}{-\cos \alpha} = -\tan \alpha $$
Thus, the cotangent of α + 3π/2 is equal to the negative of the tangent of α.
Example: Calculating sin(330°)
Let us use these identities to evaluate:
$$ \sin 330^\circ $$
First, rewrite 330° as:
$$ 330^\circ = 270^\circ + 60^\circ $$
Therefore:
$$ \sin 330^\circ = \sin(270^\circ + 60^\circ) $$
In radians:
$$ \sin \left( \frac{3\pi}{2} + \frac{\pi}{3} \right) $$
Since α = π/3, we can apply the identity:
$$ \sin \left( \alpha + \frac{3\pi}{2} \right) = -\cos \alpha $$
Substituting α = π/3:
$$ \sin \left( \frac{3\pi}{2} + \frac{\pi}{3} \right) = -\cos \left( \frac{\pi}{3} \right) $$
Because
$$ \cos \left( \frac{\pi}{3} \right) = \frac{1}{2}, $$
we obtain:
$$ \sin \left( \frac{3\pi}{2} + \frac{\pi}{3} \right) = -\frac{1}{2}. $$
Hence:
$$ \sin 330^\circ = -\frac{1}{2}. $$
Instead of evaluating the sine of a fourth-quadrant angle directly, we transformed the problem into the simpler calculation of -cos(60°).
Final Remarks
The identities for α + 3π/2 are part of a broader set of trigonometric reduction formulas used to relate angles outside the first quadrant to simpler reference angles. Mastering these relationships can significantly speed up calculations and provide a deeper understanding of how trigonometric functions behave on the unit circle.
