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Finding a Side in a Right Triangle Using Trigonometry — Definition, Formula & Examples

Finding a side in a right triangle using trigonometry means using sine, cosine, or tangent ratios along with a known angle and one known side to calculate an unknown side length.

Given a right triangle with one acute angle θ\theta, a known side length, and an unknown side length, the unknown side can be determined by selecting the appropriate trigonometric ratio — sin⁡θ\sin\theta, cos⁡θ\cos\theta, or tan⁡θ\tan\theta — that relates the known side, the unknown side, and the given angle, then solving the resulting equation algebraically.

Key Formula

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}
Where:
  • θ\theta = A known acute angle of the right triangle
  • opposite\text{opposite} = The side across from angle θ
  • adjacent\text{adjacent} = The side next to angle θ (not the hypotenuse)
  • hypotenuse\text{hypotenuse} = The longest side, opposite the right angle

How It Works

Start by labeling the sides of the right triangle relative to the known acute angle: the side directly across from the angle is the **opposite**, the side next to the angle (that is not the hypotenuse) is the **adjacent**, and the longest side across from the right angle is the **hypotenuse**. Use the mnemonic SOH CAH TOA to pick the correct ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Write the equation using the ratio that involves both the known side and the unknown side, then rearrange to isolate the unknown. Finally, evaluate with a calculator (set to degree or radian mode as needed).

Worked Example

Problem: In a right triangle, one acute angle is 35° and the hypotenuse is 10 cm. Find the length of the side opposite the 35° angle.
Label the sides: Relative to the 35° angle, the unknown side is the opposite and the known side (10 cm) is the hypotenuse.
Choose the ratio: Sine connects opposite and hypotenuse (SOH).
sin⁡35°=opposite10\sin 35° = \frac{\text{opposite}}{10}
Solve for the unknown: Multiply both sides by 10 to isolate the opposite side.
opposite=10×sin⁡35°\text{opposite} = 10 \times \sin 35°
Calculate: Use a calculator in degree mode: sin 35° ≈ 0.5736.
opposite≈10×0.5736=5.74 cm\text{opposite} \approx 10 \times 0.5736 = 5.74 \text{ cm}
Answer: The side opposite the 35° angle is approximately 5.74 cm.

Another Example

Problem: A right triangle has an acute angle of 50° and the side adjacent to that angle is 8 m. Find the length of the side opposite the 50° angle.
Label the sides: The known side (8 m) is adjacent to the 50° angle. The unknown side is opposite the 50° angle.
Choose the ratio: Tangent connects opposite and adjacent (TOA).
tan⁡50°=opposite8\tan 50° = \frac{\text{opposite}}{8}
Solve for the unknown: Multiply both sides by 8.
opposite=8×tan⁡50°\text{opposite} = 8 \times \tan 50°
Calculate: tan 50° ≈ 1.1918.
opposite≈8×1.1918=9.53 m\text{opposite} \approx 8 \times 1.1918 = 9.53 \text{ m}
Answer: The side opposite the 50° angle is approximately 9.53 m.

Why It Matters

This skill appears repeatedly in high school geometry and trigonometry courses, and it is tested on the SAT and ACT. Engineers, architects, and surveyors use it constantly — for example, calculating the height of a building from a measured distance and angle of elevation. Mastering this technique also builds the foundation for the Law of Sines and Law of Cosines, which extend trigonometry to non-right triangles.

Common Mistakes

Mistake: Using the wrong trig ratio, such as using sine when tangent is needed.
Correction: Always label the sides as opposite, adjacent, and hypotenuse relative to the given angle first. Then match the pair of sides (one known, one unknown) to the correct SOH CAH TOA ratio.
Mistake: Having the calculator set to radian mode when the angle is given in degrees.
Correction: Check your calculator's mode before computing. An angle of 35° in radian mode gives a completely wrong answer (sin 35 radians ≈ −0.4288 instead of sin 35° ≈ 0.5736).

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