Relationships Between Trigonometric Functions of the Same Argument
Big Idea: All six trigonometric functions — $\sin\alpha$, $\cos\alpha$, $\tan\alpha$, $\cot\alpha$, $\sec\alpha$, and $\csc\alpha$ — are interconnected. If you know the value of one trigonometric function and the quadrant of the angle, you can find all the others.
1. The Fundamental (Pythagorean) Identity
The Most Important Identity
$$\sin^2\alpha + \cos^2\alpha = 1$$
This comes directly from the Pythagorean theorem applied to the unit circle. For any point $(x, y)$ on the unit circle, $x = \cos\alpha$ and $y = \sin\alpha$, and since $x^2 + y^2 = 1$, we get this identity.
Derived Pythagorean Identities:
Dividing the fundamental identity by $\cos^2\alpha$:
$$\tan^2\alpha + 1 = \sec^2\alpha$$
Dividing the fundamental identity by $\sin^2\alpha$:
$$1 + \cot^2\alpha = \csc^2\alpha$$
Unit Circle Visualization
Interactive visualization — available in the full lesson