Trigonometric Identities for the Angles α and π + α
In trigonometry, the angles α and π + α satisfy the following identities: $$ \sin(\pi+\alpha) = -\sin(\alpha) $$ $$ \cos(\pi+\alpha) = -\cos(\alpha) $$ $$ \tan(\pi+\alpha) = \tan(\alpha) $$ $$ \cot(\pi+\alpha) = \cot(\alpha) $$
These identities are particularly useful when simplifying trigonometric expressions and evaluating the sine, cosine, tangent, or cotangent of angles greater than 180°.
The reason they work is simple: the angles α and π + α differ by π radians (180°), which places them at opposite points on the unit circle.
Why Do These Identities Hold?
Consider an angle α and the angle π + α on the unit circle.
The difference between the two angles is exactly π radians:
$$ (\pi + \alpha) - \alpha = \pi $$
In other words, starting from α, we rotate an additional half-turn around the circle to reach π + α.
Now construct the right triangles OAB and OCD.
The triangles are congruent because they have the same hypotenuse, the same acute angle α, and a right angle.
As a result, their corresponding sides have equal lengths.
Sine
The vertical legs of the two triangles are equal in length, so the sine values have the same absolute value:
$$ |\sin(\pi+\alpha)| = |\sin\alpha| $$
The diagram below shows what happens geometrically.
The point associated with α lies above the x-axis, while the point associated with π + α lies below it.
Therefore, the sine values have opposite signs:
$$ \sin(\pi+\alpha) = -\sin(\alpha) $$
Equivalent form:
$$ \sin\alpha = -\sin(\pi+\alpha) $$
Cosine
The same idea applies to the cosine function. Since the horizontal legs of the two triangles are equal, the cosine values also have the same absolute value.
However, the point corresponding to α lies on the positive side of the x-axis, whereas the point corresponding to π + α lies on the negative side.
Consequently, the cosine values have opposite signs:
$$ \cos(\pi+\alpha) = -\cos(\alpha) $$
Equivalent form:
$$ \cos\alpha = -\cos(\pi+\alpha) $$
Tangent
The tangent is defined as the ratio of sine to cosine:
$$ \tan(\pi+\alpha) = \frac{\sin(\pi+\alpha)}{\cos(\pi+\alpha)} $$
Substituting the identities for sine and cosine gives:
$$ \tan(\pi+\alpha) = \frac{-\sin\alpha}{-\cos\alpha} = \tan\alpha $$
Therefore, the tangent does not change when π radians are added to the angle.
Cotangent
The cotangent is defined as the ratio of cosine to sine:
$$ \cot(\pi+\alpha) = \frac{\cos(\pi+\alpha)}{\sin(\pi+\alpha)} $$
Substituting the corresponding identities yields:
$$ \cot(\pi+\alpha) = \frac{-\cos\alpha}{-\sin\alpha} = \cot\alpha $$
Thus, the cotangent also remains unchanged.
To summarize:
- Sine changes sign.
- Cosine changes sign.
- Tangent remains unchanged.
- Cotangent remains unchanged.
Whenever two angles differ by π radians, these relationships always hold.
A Practical Example
Let us calculate the sine of 210°.
$$ \sin 210^\circ $$
First, rewrite 210° as:
$$ 210^\circ = 180^\circ + 30^\circ $$
Therefore:
$$ \sin 210^\circ = \sin(180^\circ + 30^\circ) $$
Expressed in radians:
$$ \sin 210^\circ = \sin\left(\pi + \frac{\pi}{6}\right) $$
Since α = π/6, we can apply the identity
$$ \sin(\pi+\alpha) = -\sin(\alpha) $$
to obtain:
$$ \sin 210^\circ = -\sin\left(\frac{\pi}{6}\right) $$
Because
$$ \sin 30^\circ = \frac{1}{2} $$
we get:
$$ \sin 210^\circ = -\frac{1}{2} $$
This approach makes it easy to evaluate trigonometric functions of many angles by reducing them to the corresponding values of simpler, well-known angles.
