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Finding Angles in a Right Triangle — Definition, Formula & Examples

Finding angles in a right triangle means using the known side lengths and inverse trigonometric functions to calculate the unknown acute angles. Given any two sides of a right triangle, you can determine either non-right angle with arcsin, arccos, or arctan.

In a right triangle with legs aa and bb and hypotenuse cc, an acute angle θ\theta satisfies sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, and tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}. The angle is recovered by applying the corresponding inverse function: θ=sin⁡−1 ⁣(oppositehypotenuse)\theta = \sin^{-1}\!\left(\frac{\text{opposite}}{\text{hypotenuse}}\right), θ=cos⁡−1 ⁣(adjacenthypotenuse)\theta = \cos^{-1}\!\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right), or θ=tan⁡−1 ⁣(oppositeadjacent)\theta = \tan^{-1}\!\left(\frac{\text{opposite}}{\text{adjacent}}\right).

Key Formula

θ=tan⁡−1 ⁣(oppositeadjacent)\theta = \tan^{-1}\!\left(\frac{\text{opposite}}{\text{adjacent}}\right)
Where:
  • θ\theta = The unknown acute angle you are solving for
  • opposite\text{opposite} = Length of the side across from angle θ
  • adjacent\text{adjacent} = Length of the side next to angle θ (not the hypotenuse)

How It Works

Start by identifying which two sides you know relative to the angle you want. Label them as opposite, adjacent, or hypotenuse with respect to that angle. Choose the trig ratio that uses exactly those two sides — sin if you have opposite and hypotenuse, cos if you have adjacent and hypotenuse, tan if you have opposite and adjacent. Then apply the matching inverse function on your calculator (make sure it is set to degrees if you want degrees). Remember that the two acute angles in a right triangle always sum to 90°, so once you find one, subtract from 90° to get the other.

Worked Example

Problem: A right triangle has legs of length 3 and 4. Find the angle opposite the side of length 3.
Step 1: Label the sides: Relative to the angle θ opposite the side of length 3: opposite = 3, adjacent = 4, hypotenuse = 5.
Step 2: Choose the right ratio: You know the opposite and adjacent sides, so use tangent.
tan⁡θ=34=0.75\tan\theta = \frac{3}{4} = 0.75
Step 3: Apply the inverse function: Use arctan (calculator in degree mode).
θ=tan⁡−1(0.75)≈36.87°\theta = \tan^{-1}(0.75) \approx 36.87°
Step 4: Find the other acute angle: The two acute angles sum to 90°.
α=90°−36.87°=53.13°\alpha = 90° - 36.87° = 53.13°
Answer: The angle opposite the side of length 3 is approximately 36.87°, and the other acute angle is approximately 53.13°.

Another Example

Problem: A right triangle has a hypotenuse of 10 and one leg of length 6. Find the angle adjacent to the leg of length 6.
Step 1: Identify sides relative to the angle: Adjacent = 6, hypotenuse = 10. Use cosine.
cos⁡θ=610=0.6\cos\theta = \frac{6}{10} = 0.6
Step 2: Apply arccos: Evaluate the inverse cosine.
θ=cos⁡−1(0.6)≈53.13°\theta = \cos^{-1}(0.6) \approx 53.13°
Answer: The angle is approximately 53.13°.

Why It Matters

This skill appears constantly in high-school geometry and trigonometry courses, as well as on the SAT and ACT. Engineers, surveyors, and architects use it to convert measured distances into angles for construction layouts and land boundaries. It also forms the foundation for solving oblique triangles later with the Law of Sines and Law of Cosines.

Common Mistakes

Mistake: Using the wrong calculator mode (radians instead of degrees).
Correction: Check that your calculator is set to degree mode before evaluating an inverse trig function, unless the problem specifically asks for radians.
Mistake: Mixing up which side is opposite and which is adjacent relative to the target angle.
Correction: Always label sides from the perspective of the specific angle you are finding. The opposite side is directly across the triangle from that angle; the adjacent side touches the angle but is not the hypotenuse.

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