Sine in terms of Secant

Theorem
Let $\theta$ be an angle.

Then:

where $\sin$ denotes the sine function and $\sec$ denotes the secant function.

Proof
We also have that:
 * In quadrant I, and quadrant II, $\sin \theta > 0$
 * In quadrant III, and quadrant IV, $\sin \theta < 0$
 * In quadrant I, and quadrant IV, $\sec \theta > 0$
 * In quadrant II, and quadrant III, $\sec \theta < 0$.

Therefore:
 * In quadrant I, $\sin \theta > 0$ and $\sec \theta > 0$.
 * In quadrant II, $\sin \theta > 0$ and $\sec \theta < 0$.
 * In quadrant III, $\sin \theta < 0$ and $\sec \theta < 0$.
 * In quadrant IV, $\sin \theta < 0$ and $\sec \theta > 0$.

Also see

 * Trigonometric Functions in terms of each other