Trig Identities

^2θ + ^2θ = 1 — and the even/odd symmetries.

The idea

An identity is an equation that holds for every value of its variable — a fact about the functions themselves, not an equation to solve. Trigonometric identities rewrite an expression into an equivalent form that is easier to work with. The most used one comes from the unit circle.

Theorem (Pythagorean identity).

For every angle $\theta$, $\sin^{2}\theta + \cos^{2}\theta = 1.$

The point at angle $\theta$ is $(\cos\theta, \sin\theta)$, at distance $1$ from the origin, and the identity is the Pythagorean theorem applied to its horizontal and vertical distances. The identity determines the size of each function from the other, since knowing one leg of a right triangle with hypotenuse $1$ fixes the other leg. It cannot determine the sign, because squaring discards signs; the quadrant of $\theta$ supplies that. It also bounds both functions, since neither square can exceed $1$.

A second pair of identities comes from the circle's symmetry. Replacing $\theta$ by $-\theta$ reflects the point across the $x$-axis, which keeps the first coordinate and flips the sign of the second.

Proposition (Even and odd identities).

For every angle $\theta$, $\cos(-\theta) = \cos\theta, \qquad \sin(-\theta) = -\sin\theta.$

In the language of functions, cosine is even and sine is odd.

Ways to work on it

• Walkthrough. ^2 + ^2 = 1 via the unit circle.
• Practice. (-θ) = θ, (-θ) = - θ.
• Hardest. From a given θ or θ, find an exact value at -θ.

Not sure where to start? Take the ten-question placement test.