SL 3.6—Trigonometric identities

Transform identities instead of guessing angles.

The identity sin²θ+cos²θ=1 and double-angle formulas rewrite expressions without solving for θ.

Worked example

If sinθ=3/5 and θ is acute, cos2θ=1−2sin²θ=7/25.

Choose an identity that matches the information already given.

An identity is true for every allowed angle, not a numerical approximation.

Complete double-angle set: $\sin2\theta=2\sin\theta\cos\theta$ and $\cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1$. If $\sin\theta=3/5$ and $\theta$ is acute, then $\cos\theta=4/5$, so $\sin2\theta=24/25$ and $\cos2\theta=7/25$. Without a quadrant condition, the sign of the missing ratio may not be unique.