How can I use inverse sine, inverse cosine, and inverse tangent to find an angle?

First label the sides relative to the angle you are looking for: opposite, adjacent, and hypotenuse. Then choose the ratio that uses the sides you know:

1. Opposite ÷ hypotenuse: use sin⁻¹, also called arcsin.
2. Adjacent ÷ hypotenuse: use cos⁻¹, or arccos.
3. Opposite ÷ adjacent: use tan⁻¹, or arctan.

For example, if opposite ÷ hypotenuse is 0.5, enter sin⁻¹(0.5). The principal result is 30° or π/6 radians. [S1]

Think of the inverse function as answering: “What angle has this sine, cosine, or tangent?” It takes a ratio as input and gives an angle as output.

Set the calculator to the units your answer should use before interpreting the result. Look for a DEG or RAD indicator; then use the inverse key, often labeled sin⁻¹, cos⁻¹, or tan⁻¹. On some calculators, you press a SHIFT or 2nd key before the usual trig key. Enter the ratio in parentheses if the calculator allows it.

For instance, to find the angle when adjacent ÷ hypotenuse is 0.5, enter cos⁻¹(0.5). You get 60° in degree mode, or π/3 in radian mode. Check the display’s mode rather than assuming the number is degrees. Calculators can display results in degrees or radians, so confirming the mode matters. [S1]

A useful check is to apply the original function to your answer: if the ratio was 0.5 for cosine, then cos(60°) should be 0.5. This checks the calculation, but it does not change which output range the inverse function uses.

Here are three calculator-ready examples. The values are chosen to give familiar exact angles:

1. If opposite ÷ hypotenuse = 1/2, calculate sin⁻¹(1/2) = 30° (π/6 radians). In other words, the sine of 30° is 1/2. [S1]
2. For a general cosine input of −√3/2, calculate cos⁻¹(−√3/2) = 150° (5π/6 radians). This is an inverse-trig example, not a ratio of positive side lengths in a right triangle. [S1]
3. If opposite ÷ adjacent = 1, calculate tan⁻¹(1) = 45° (π/4 radians). [S1]

In sin⁻¹(x), the “−1” means inverse function, not “raise to the power of negative one.” It does not mean 1/sin(x). The reciprocal of sine is cosecant, a different function. [S1] A quick way to keep them apart: sin⁻¹(0.5) asks for an angle; 1/sin(0.5) asks for a reciprocal value, with the input interpreted according to your calculator’s mode.

Not necessarily. Inverse sine, cosine, and tangent return angles in defined principal ranges: arcsin from −π/2 to π/2, arccos from 0 to π, and arctan strictly between −π/2 and π/2. [S1] Those ranges identify which one angle the inverse function returns for an input; they do not list every angle that might have the same trig ratio.

If your problem gives a diagram or restricts the angle to a particular interval, compare the calculator result with that information. In a right triangle, use the diagram and the fact that its acute angles must fit the triangle; do not accept an output just because the calculator produced it. If the question asks for other angles beyond the inverse function’s principal result, use the problem’s interval and trig relationships to determine which additional angles qualify. The cited conventions establish principal ranges, not every solution for every possible interval.

Before submitting, write down the ratio you used, the inverse function, the calculator mode, and the resulting angle. That makes a wrong side choice or degree-radian mismatch easier to spot.