Angles α and 2π - α in Trigonometry
In trigonometry, the angles α and 2π - α are closely related. Their positions on the unit circle are symmetric with respect to the x-axis, which leads to a simple set of reduction formulas: $$ \sin(2 \pi - \alpha) = -\sin(\alpha) $$ $$ \cos(2 \pi - \alpha) = \cos(\alpha) $$ $$ \tan(2 \pi - \alpha) = -\tan(\alpha) $$ $$ \cot(2 \pi - \alpha) = -\cot(\alpha) $$
These identities make it easy to evaluate trigonometric functions of angles greater than 180° by rewriting them in terms of a familiar reference angle.
Why Do These Formulas Work?
To understand these identities, place the angles α and 2π - α on the unit circle.
The angle 2π - α is obtained by rotating clockwise from the positive x-axis by an amount α. As a result, its terminal side is the mirror image of the terminal side of α across the x-axis.
On the unit circle, the point corresponding to α has coordinates
$$ (\cos\alpha,\sin\alpha) $$
while the point corresponding to 2π - α has coordinates
$$ (\cos\alpha,-\sin\alpha) $$
Notice what changes:
- The x-coordinate remains the same.
- The y-coordinate changes sign.
Since cosine corresponds to the x-coordinate and sine corresponds to the y-coordinate, we immediately obtain:
$$ \cos(2\pi-\alpha)=\cos\alpha $$
$$ \sin(2\pi-\alpha)=-\sin\alpha $$
The formulas for tangent and cotangent follow directly from their definitions:
$$ \tan(2\pi-\alpha)=-\tan\alpha $$
$$ \cot(2\pi-\alpha)=-\cot\alpha $$
This is exactly the same behavior observed for the angles α and -α because 2π - α and -α determine the same terminal side. In other words, they are coterminal angles.
Example: Calculating sin 330°
Let's use the formula to compute the sine of 330°.
$$ \sin 330^\circ $$
First, rewrite 330° as 360° - 30°:
$$ \sin 330^\circ = \sin(360^\circ - 30^\circ) $$
In radians, this becomes:
$$ \sin 330^\circ = \sin\left(2\pi-\frac{\pi}{6}\right) $$
Since the angle has the form 2π - α, with
$$ \alpha=\frac{\pi}{6} $$
we can apply the reduction formula:
$$ \sin(2\pi-\alpha)=-\sin\alpha $$
Therefore,
$$ \sin 330^\circ =\sin\left(2\pi-\frac{\pi}{6}\right) =-\sin\left(\frac{\pi}{6}\right) $$
The sine of 30° (or π/6 radians) is a well-known value:
$$ \sin\left(\frac{\pi}{6}\right)=\frac{1}{2} $$
Substituting this value gives:
$$ \sin 330^\circ=-\frac{1}{2} $$
So the complete calculation is:
$$ \sin 330^\circ =\sin\left(2\pi-\frac{\pi}{6}\right) =-\sin\left(\frac{\pi}{6}\right) =-\frac{1}{2} $$
By using reduction formulas, complicated angles can be transformed into simpler reference angles, making trigonometric calculations faster and easier to understand.
