Solving Trigonometric Equations

Sine is a vertical coordinate: two points reflected across the vertical axis have the same sine. Cosine is a horizontal coordinate: reflection across the horizontal axis preserves cosine.

Let $\alpha$ be the calculator's principal angle for the required ratio:

• Sine: use $\alpha$ and $180^\circ-\alpha$, then add or subtract 360° as needed.
• Cosine: use $\alpha$ and $360^\circ-\alpha$, then add or subtract 360°.
• Tangent: start with $\alpha$ and add or subtract 180°. Opposite points have the same coordinate ratio.

In radians, replace 180° by $\pi$ and 360° by $2\pi$. Generate only as many repeats as the interval needs; write each distinct angle once.

For $\sin x=-1/2$ on $-\pi\le x\le\pi$, inverse sine gives $-\pi/6$. The partner is $\pi-(-\pi/6)=7\pi/6$, outside the interval. Subtract $2\pi$ to get $-5\pi/6$. The answers are $-5\pi/6,-\pi/6$.

For $\cos x=1$ on $0^\circ\le x\le360^\circ$, include both 0° and 360°: they are different input angles even though they reach the same point. With $x<360^\circ$, exclude 360°.

At a maximum or minimum, the two partner rules can give the same angle; count it only once. For example, $\sin x=1$ has just 90° in one turn. If a sine or cosine ratio is outside $[-1,1]$, there is no real solution.

In RAD mode, keep special angles as exact multiples of $\pi$. Other radian answers normally use 3 significant figures unless the question specifies otherwise.