Unit Circle Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Use the unit circle to prove the Pythagorean identity

\sin^2\theta+\cos^2\theta=1

and derive

1+\tan^2\theta=\sec^2\theta

1
A point on the unit circle satisfies $x^2+y^2=1$ . Since $x=\cos\theta$ and $y=\sin\theta$ , substituting gives $\cos^2\theta+\sin^2\theta=1$ . ✓
2
Divide both sides by $\cos^2\theta$ (for $\cos\theta\neq0$ ): $1+\frac{\sin^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta}$ .
3
Use $\tan\theta=\sin\theta/\cos\theta$ and $\sec\theta=1/\cos\theta$ : $1+\tan^2\theta=\sec^2\theta$ . ✓

Answer

\sin^2\theta+\cos^2\theta=1

;

1+\tan^2\theta=\sec^2\theta

The Pythagorean identity is the equation of the unit circle itself. Dividing by

\cos^2\theta

\sin^2\theta

generates the two secondary Pythagorean identities. These are foundational tools in trigonometry.

About Unit Circle

The circle of radius 1 centered at the origin in the coordinate plane, used to define trigonometric functions for all angles.

Verify that the point [formula] lies on the unit circle and identify the angle [formula].

Find [formula], [formula], and [formula] for [formula] using the unit circle. Identify which quadran

Using the unit circle, find the exact values of [formula], [formula], and [formula] for [formula] an