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How to solve a right triangle with one side and one angle

One positive side and one acute angle determine the rest of a right triangle. Angle C is 90° and is not something you type in. The other acute angle is 90° minus the angle you know, and the sides follow from sine and cosine.

Name the sides before you calculate

Side a is opposite angle A. Side b is opposite angle B. Side c, the hypotenuse, is opposite the right angle. Mixing up opposite and adjacent is the usual reason a trig answer looks wrong.

Which ratio to use

• Known side is opposite the known angle: hypotenuse = opposite ÷ sin(angle), and the other leg = hypotenuse × cos(angle).
• Known side is adjacent to the known angle: hypotenuse = adjacent ÷ cos(angle), and the opposite leg = hypotenuse × sin(angle).
• Known side is the hypotenuse: opposite = hypotenuse × sin(angle), and adjacent = hypotenuse × cos(angle).

Those six pairings are side a or b or c with angle A or B. A 30° angle and hypotenuse 10 give an opposite side of 5. The other acute angle is 60°.

Steps

1. Mark the right angle. That is C. Do not enter 90° as the known angle.
2. Decide whether your angle is A or B, and whether your length is a, b, or c.
3. Choose sine or cosine from the list above.
4. Find the complement: 90° minus the known angle.
5. Check that every side is positive, c is the longest side, and A + B = 90°.

Common mistakes

• Entering 0°, 90°, or an angle outside that open range. None of those is an acute angle in this triangle.
• Using degree mode in your head but forgetting a calculator’s sine function may expect radians. The app converts degrees before it calls sine and cosine.
• Swapping a and b, which swaps opposite and adjacent and swaps angles A and B.

How the app does it

Switch to Advanced. Choose the known side and the known angle, enter the length and the degrees, then tap Solve Triangle. The steps show the ratio that was used. Sine, cosine, and tangent for A and B are available after the solve. If you already know two sides and no angle, stay in Basic. Finding the hypotenuse does not need trigonometry.

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