How can I solve a right triangle when one acute angle and one side are known?

Start by naming the sides in relation to your known angle θ: the hypotenuse is across from the right angle; the opposite side is across from θ; and the adjacent side touches θ but is not the hypotenuse. Then identify the side you know and the side you want.

Use the ratio that connects those two sides: sine relates opposite and hypotenuse, cosine relates adjacent and hypotenuse, and tangent relates opposite and adjacent. In symbols, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. [S1]

For instance, if you know the angle and hypotenuse but want the opposite side, use sine. If you know the angle and adjacent side and want the hypotenuse, use cosine. If your angle is given in degrees, make sure your calculator is set to degrees.

Assume the known angle is 35° and the known side is the adjacent side, 12 units. We want the opposite side, the hypotenuse, and the remaining acute angle.

1. For the opposite side, use tangent because it relates opposite and adjacent: tan 35° = opposite/12. Rearranging gives opposite = 12 tan 35° ≈ 8.40 units.
2. For the hypotenuse, use cosine because it relates adjacent and hypotenuse: cos 35° = 12/hypotenuse. Rearranging gives hypotenuse = 12/cos 35° ≈ 14.65 units.
3. The two acute angles add to 90°, so the other angle is 90° − 35° = 55°. [S1]

The complete solution, to two decimal places, is: adjacent = 12 units, opposite ≈ 8.40 units, hypotenuse ≈ 14.65 units, and the other acute angle = 55°. The decimal side lengths are approximate because the trigonometric calculations have been rounded.

First check the shape: the hypotenuse should be longer than either leg. In the example, 14.65 is greater than both 12 and 8.40. Also check that your setup matches the side labels: tangent uses opposite and adjacent, while cosine uses adjacent and hypotenuse. Small differences in the displayed decimals can result from rounding.

For a different known side, keep the same process: identify its position relative to the given angle, name the side you want, and choose the ratio containing both. Rearrange that ratio to isolate the unknown. For example, if the adjacent side is known and you need the hypotenuse, cosine gives cos θ = adjacent/hypotenuse, so hypotenuse = adjacent/cos θ. The ratios and complementary-angle relationship apply to right triangles with an acute angle. [S1]

Yes. Write down the reference angle before labeling any sides; opposite and adjacent depend on which acute angle you are using. Then fill in this template:

• Known angle: θ = [angle]°
• Known side: [length], and it is the [opposite/adjacent/hypotenuse]
• Side sought: [opposite/adjacent/hypotenuse]
• Ratio connecting those sides: [sin/cos/tan] θ = [side relationship]
• Substitution and rearrangement: [equation] = [answer with units]
• Other acute angle: 90° − θ = [answer]°
• Check: the hypotenuse is longer than either leg; confirm the ratio uses the intended sides

If the calculator result seems wrong, check the angle mode, side labels, and algebra before rounding. These checks can catch common setup or arithmetic slips, but they do not establish that a diagram or its measurements were recorded correctly.