Trigonometric Functions

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Geometric definitions of trigonometric functions for real values.

Right triangle

For angles \(0 \lt \alpha \lt \frac{\pi}{2}\), the trigonometric functions can be defined as ratios of side lengths in a right triangle.

hypotenuse adjacent opposite α
$$ \begin{align*} \sin \alpha &= \frac{\text{opposite}}{\text{hypotenuse}}\\ \cos \alpha &= \frac{\text{adjacent}}{\text{hypotenuse}}\\ \tan \alpha &= \frac{\text{opposite}}{\text{adjacent}} \end{align*} $$

Another three are obtained via reciprocals: The cosecant (\(\csc\)) from the sine, the secant (\(\sec\)) from the cosine, and the cotangent (\(\cot\)) from the tangent.

Unit circle

We can generalize these definitions at the unit circle for any angle. As the name implies, the radius of this circle has length one (unit length).

$$ r^2 = x^2 + y^2 = \cos^2 \alpha + \sin^2 \alpha = 1 $$

Thus, for any point on the circle, the \(x\) component is the cosine (green), the \(y\) component the sine (red). The tangent (dashed) is the ratio of sine to cosine.