Solving right triangles
One side and one acute angle are enough to find everything else about a right triangle. With side 7 opposite a 30° angle, the hypotenuse is exactly 14.
Finding a missing side
OpenStax puts it plainly: the real power of right-triangle trigonometry shows when you know an angle but not all the sides. The method has three steps. First pick the ratio that contains the side you want and a side you know. Then write the equation with the known angle. Finally solve for the unknown. Take OpenStax's own example: angle 30°, the opposite side 7, and the adjacent side a unknown. Tangent links opposite and adjacent, so tan 30° = 7/a, which rearranges to a = 7 / tan 30° ≈ 12.1. For the hypotenuse c, sine links opposite and hypotenuse: sin 30° = 7/c, so c = 7 / sin 30° = 14.
Watch where the unknown sits. If it is on top of the fraction you multiply: a ladder 5 m long leaning at 75° to the ground reaches a height of 5 sin 75° ≈ 4.83 m, and its foot stands 5 cos 75° ≈ 1.29 m from the wall. If the unknown is underneath, you divide, as with the 7 / tan 30° above. A quick check: the hypotenuse must come out longer than both legs, and the side opposite the bigger acute angle must be the longer leg.
Finding a missing angle
Sometimes you know two sides and want the angle. Then you run a ratio backwards with an inverse function, written sin⁻¹, cos⁻¹ and tan⁻¹ (also called arcsine, arccosine and arctangent). If the opposite side is 3 and the adjacent side is 4, then tan θ = 3/4 = 0.75 and θ = tan⁻¹(0.75) ≈ 36.87°. The other acute angle is 90 − 36.87 = 53.13°. The −1 here is a name, not a power: sin⁻¹ x means "the angle whose sine is x", not 1 / sin x.
Drag to turn. Scroll or pinch to zoom.
Solving the whole triangle
To solve a right triangle means to find all three sides and both acute angles. From two sides, use the Pythagorean theorem for the third and an inverse function for an angle; the last angle is 90° minus the first. From one side and one angle, the other acute angle is 90° minus the given one, and two ratios give the remaining sides. With legs 6 and 8, for example, the hypotenuse is √(36 + 64) = 10, the angle opposite 6 is tan⁻¹(6/8) ≈ 36.87° and the other angle is about 53.13°. It is the 3-4-5 triangle scaled by 2, so the angles match.
It helps to check an answer with a second ratio. In the OpenStax example the sides came out as 7, about 12.12 and 14. Then cos 30° should equal 12.12 / 14 ≈ 0.866, and it does: cos 30° = √3/2 ≈ 0.866. A mismatch means one of the sides was found with the wrong ratio.
- Angle 30°, opposite side 7. Adjacent side to 2 decimal places?
- Same triangle. Hypotenuse?
- A 5 m ladder makes 75° with the ground. How high up the wall does it reach, to 2 decimal places?
- Same ladder. Distance of its foot from the wall, to 2 decimal places?
- Opposite 3, adjacent 4. θ in degrees to 2 decimal places?
- Opposite 5, hypotenuse 10. θ?
- A ramp rises 0.5 m over a horizontal run of 6 m. Its angle to 2 decimal places?
- Legs 6 and 8. Hypotenuse?
- Hypotenuse 12, angle 40°. Side adjacent to 40°, to 2 decimal places?
- What does sin⁻¹(0.6) mean?
- the angle whose sine is 0.6
- 1 ÷ sin 0.6
- sin 0.6 squared
- −sin 0.6
- You find a hypotenuse of 4 and a leg of 5. What went wrong?
- the hypotenuse must be the longest side
- nothing
- the angle must be 45°
- legs are always shorter than 1
- Pick the ratio that holds the unknown side and a known side.
- Unknown on top: multiply. Unknown below: divide.
- Inverse functions sin⁻¹, cos⁻¹, tan⁻¹ turn a ratio back into an angle.
- The two acute angles add to 90°; use Pythagoras for a third side.
- Round only at the end.