Finding Missing Angles: Step-by-Step Guide
In a right triangle, if you know any two sides, you can find an unknown acute angle. Pick the trig ratio (sin, cos, or tan) that uses the two sides you know, then use the matching inverse function (arcsin, arccos, or arctan) to get the angle.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Label the sides. From the point of view of the angle you want, label the sides opposite, adjacent, and hypotenuse.\[\text{opposite},\;\text{adjacent},\;\text{hypotenuse}\]
- Pick the ratio (SOH-CAH-TOA). Choose sin, cos, or tan, depending on which two sides you know.\[\sin\theta = \frac{O}{H},\;\cos\theta = \frac{A}{H},\;\tan\theta = \frac{O}{A}\]
- Write the equation. Put the known side lengths into the ratio.\[\tan\theta = \frac{O}{A}\]
- Use the inverse function. Use arcsin, arccos, or arctan on your calculator to find the angle.\[\theta = \tan^{-1}\!\left(\frac{O}{A}\right)\]
- Check. The angle should be between 0 and 90 degrees, and the sides should fit the triangle.\[0^\circ < \theta < 90^\circ\]
Worked example
In a right triangle, the side opposite angle theta is 5 and the adjacent side is 12. Find theta.
Step 1: The opposite side is 5 and the adjacent side is 12. (The hypotenuse would be the square root of (25 + 144) = 13.)
Step 2: We know the opposite and adjacent sides, so use tangent.
Step 3: tan(theta) = 5 / 12.
Step 4: theta = arctan(5 / 12), which is about 22.62 degrees.
Step 5: Check: 22.62 degrees is between 0 and 90. Also sin(22.62 degrees) is about 0.385, which is 5/13. It fits the triangle.
The angle is about 22.62 degrees (or 0.395 radians).
Tips and common mistakes
- Use SOH-CAH-TOA: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.
- Check that your calculator is in the right mode (degrees or radians).
- Double-check by finding the third side with the Pythagorean theorem and testing another ratio.
- For a triangle, you want a positive acute angle. arcsin and arctan give angles between -pi/2 and pi/2, and arccos gives angles between 0 and pi.
- If you know the hypotenuse and one leg, use sin or cos. If you know both legs, use tan.
- The two non-right angles add up to 90 degrees. Once you find one, subtract from 90 to find the other.
- Round only at the very end.
- Side lengths can be in any unit. The angle is not measured in that unit.