Inverse Trigonometric Functions

Sine takes an angle and gives a ratio. Inverse sine takes a ratio and returns an angle. Since $\sin30^\circ=1/2$, we have $\sin^{-1}(1/2)=30^\circ$.

But 150° has the same sine. To give just one answer, an inverse function uses a restricted part of the original graph. The chosen answer is called the principal value.

• Inverse sine accepts inputs from −1 to 1 and returns angles from −90° to 90°, inclusive.
• Inverse cosine accepts inputs from −1 to 1 and returns angles from 0° to 180°, inclusive.
• Inverse tangent accepts any real input and returns an angle strictly between −90° and 90°. The endpoints are excluded because tangent is undefined there.

In radians, 90° is $\pi/2$ and 180° is $\pi$. MF19 supplies these principal ranges. Match the calculator's DEG or RAD mode to the requested angle unit.

To find $\cos^{-1}(-1/2)$, cosine is negative in II and III, but only II lies in the inverse-cosine range:

Do not add 240° to an inverse-function value. You would include it when solving a cosine equation over a full turn.

To undo an inverse key, apply the original function. For instance:

So $x=-1/2$. First check that the requested angle lies in the principal range. An equation such as $\sin^{-1}x=120^\circ$ has no real solution, even though sine can be evaluated at 120°.