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  • Trigonometry in the Right Triangle

How do you find an angle when two sides are known?

Quick answer

To find an angle when two sides are known, divide the two side lengths to find their ratio, determine whether that ratio is sine, cosine, or tangent relative to the target angle, and then apply the inverse function (sin⁻¹, cos⁻¹, or tan⁻¹) on your calculator to output the angle in degrees.

The topic explained

Choose the ratio and reverse the direction

To find an angle when two sides are known in a right triangle, you run the standard trigonometry steps in reverse. When finding side lengths, you enter an angle into sine, cosine, or tangent to calculate a side ratio. Here, you compute the numerical ratio of the two sides first, and then use an inverse function to convert that ratio back into its angle.

Label your two known sides relative to the angle you need to find. If you have the opposite side and the hypotenuse, set up the sine ratio. If you have the adjacent side and the hypotenuse, use the cosine ratio. If you have both legs, use the tangent ratio.

Using the inverse keys on your calculator

The buttons labeled sin⁻¹, cos⁻¹, and tan⁻¹ are inverse trigonometric functions, often accessed by pressing the Shift or 2nd key. The minus-one notation means inverse, not a reciprocal. It does not divide one by the function; it accepts a ratio and returns an angle.

Always calculate the division before applying the function. For example, if the leg opposite your angle is 3 and the hypotenuse is 5, divide 3 by 5 to get 0.6. Enter sin⁻¹(0.6) into your calculator, which gives approximately 36.9 degrees. If both legs equal 4, divide 4 by 4 to get 1, and calculate tan⁻¹(1) to find 45 degrees.

Checking your results

Every acute angle in a right triangle must measure between 0 and 90 degrees. If your calculator returns a small decimal like 0.64 instead of 36.9, your calculator is set to radians. Switch the mode setting back to degrees.

You can check your work using the remaining angles. The two acute angles in any right triangle must add up to 90 degrees so that all three angles total 180 degrees. If one angle is 36.9 degrees, the other must be 90 minus 36.9, which equals 53.1 degrees.

Free printable worksheet

Finding Angles from Two Sides in Right Triangles

For 10th grade geometry students. For each right triangle, identify the correct ratio, calculate the unknown angle to the nearest tenth of a degree, and find the missing acute angle.

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Finding Angles from Two Sides in Right TrianglesName:

  1. Opposite side = 3, Hypotenuse = 5. Find the angle opposite the side of length 3.

  2. Adjacent side = 8, Hypotenuse = 10. Find the angle adjacent to the side of length 8.

  3. Opposite side = 4, Adjacent side = 4. Find the angle opposite the first leg.

  4. Opposite side = 5, Hypotenuse = 13. Find the angle opposite the side of length 5.

  5. Opposite side = 7, Adjacent side = 12. Find the angle opposite the side of length 7.

  6. In right triangle ABC, angle C is 90°, side AC (adjacent to A) = 9, and hypotenuse AB = 15. Find angle A and angle B.

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Common questions

Does the minus-one exponent on sin⁻¹ mean one over sine?

No. In trigonometry, sin⁻¹ denotes the inverse sine function, which takes a side ratio and outputs an angle. The expression 1/sin(x) is the cosecant function, which produces a completely different value.

Why did my calculator return an answer under one degree?

Your calculator is almost certainly in radian mode instead of degree mode. Look for a DEG/RAD toggle button or open your calculator settings to select degrees.

Why does my calculator give an error when evaluating an inverse sine?

The sine and cosine of an acute angle in a right triangle must always stay between 0 and 1, because the legs can never exceed the hypotenuse. If you entered a ratio greater than 1, you likely swapped the numerator and denominator.

How do I choose between inverse sine, cosine, and tangent?

Look at which two sides are known relative to your unknown angle. Use sine for opposite and hypotenuse, cosine for adjacent and hypotenuse, and tangent for opposite and adjacent.

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