Reading on the trigonometric circle

Elementary equations have an immediate geometric interpretation on the trigonometric circle: it makes it plain why $\sin x = k$ yields two families of solutions whereas $\tan x = k$ yields only one.

Observation — Geometric method: reading on the circle

The solutions of $\sin x = k$ are read off by drawing on the trigonometric circle the horizontal line $y=k$: its intersections with the circle are $P_{x_0}$ and $P_{\pi - x_0}$ (the reflection of $P_{x_0}$ across the $y$-axis), hence the double family. Likewise, $\cos x = k$ corresponds to the vertical line $x=k$ and gives the points $P_{x_0}$ and $P_{-x_0}$.

The solutions of $\sin x = k$: the line $y=k$ intersects the circle at the points $P_{x_0}$ and $P_{\pi-x_0}$.

Links

Topics: Trigonometric equations
Concepts: Trigonometric circle · Elementary equation
Functions: Cosine · Sine
Skills: Interpreting a graph · Solving equations