Work backwards to find an angle

1 · Learn

When two sides of a right triangle are known, form the ratio for the chosen acute angle. To recover the angle from its ratio, use the matching inverse function: arcsin, arccos or arctan. A calculator may label these sin⁻¹, cos⁻¹ and tan⁻¹. Here inverse means undoing the function, NOT taking 1 divided by the ratio.

If opposite = 3 cm and hypotenuse = 5 cm, sin(theta) = 3/5 = 0.6. In DEG mode, theta = arcsin(0.6), approximately 36.87°. Check by calculating sin(36.87°), which is close to 0.6. The other acute angle is 90° - theta, approximately 53.13°.

The same triangle has adjacent = 4 cm, so arctan(3/4) gives the same theta. Keep track of which acute angle you named: switching angles swaps opposite and adjacent. The hypotenuse remains opposite the right angle.

For an acute angle, sine and cosine ratios lie strictly between 0 and 1. A claimed sine ratio of 5/3 is impossible here: check side labels instead of forcing a calculator answer. Tangent can exceed 1, for example when opposite is longer than adjacent.

2 · Worked example

A right triangle has opposite 5 cm and adjacent 12 cm relative to theta. tan(theta) = 5/12. Keep the fraction in the calculator: theta = arctan(5/12), approximately 22.62°. Check that tan(22.62°) is close to 5/12, and that theta is acute.

3 · Your turn

For a chosen acute angle, adjacent is 12 cm and hypotenuse is 13 cm. Find the angle using cosine. Separately, opposite is 24 cm and adjacent is 7 cm: find that angle using tangent. Give two decimal places.

4 · Apply your learning

Draw and label a 3-4-5 right triangle twice. Choose a different acute angle on each copy, calculate both angles using ratios and check that their sum is approximately 90°. Explain why sin⁻¹(0.6) is an angle, while 1/0.6 is a number, not the same operation.