Pythagorist
Trigonometry

Trigonometry Calculator

Use SOHCAHTOA to find missing sides and angles in right-angled triangles. Select the ratio, choose what you want to find, and get the answer with clear working.

Opposite side

5

How it works

  1. Identify the ratio: SOHCAHTOA — Sine

    sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

    Using the sin ratio: sintheta = frac{text{opposite}}{text{hypotenuse}}.

  2. Rearrange and calculate

    opp=sin⁡(30°)×10=5\text{opp} = \sin(30°) \times 10 = 5

    Result = 5.

    Result:5

SOHCAHTOA

sin⁡θ=OH,cos⁡θ=AH,tan⁡θ=OA\sin\theta = \frac{O}{H}, \quad \cos\theta = \frac{A}{H}, \quad \tan\theta = \frac{O}{A}
OOOpposite — the side opposite the angle θ
AAAdjacent — the side next to the angle θ
HHHypotenuse — the longest side
θθThe angle in degrees

Example substitution

opp=sin⁡(30°)×10=0.5×10=5\text{opp} = \sin(30°) \times 10 = 0.5 \times 10 = 5

With a 30° angle and hypotenuse of 10, the opposite side = 5.

Worked examples

A right-angled triangle has an angle of 30° and a hypotenuse of 14 cm. Find the opposite side.

  1. 1
    Identify ratio: sin(θ) = opposite / hypotenuse (SOH)
  2. 2
    Substitute: opposite = sin(30°) × 14
  3. 3
    Calculate: opposite = 0.5 × 14
Answer: Opposite side = 7 cm

A triangle has an adjacent side of 8 m and hypotenuse of 10 m. Find the angle.

  1. 1
    Identify ratio: cos(θ) = adjacent / hypotenuse (CAH)
  2. 2
    Rearrange: θ = cos⁻¹(adjacent / hypotenuse)
  3. 3
    Substitute: θ = cos⁻¹(8 / 10) = cos⁻¹(0.8)
Answer: θ = 36.87°

Frequently asked questions

How do you use trigonometry to find a missing side?+

Identify which sides are involved (opposite, adjacent, hypotenuse) relative to the angle, choose the correct ratio (SOH, CAH, or TOA), then solve.

How do you use trigonometry to find a missing angle?+

Set up the ratio with the known sides, then use the inverse trig function (sin⁻¹, cos⁻¹, or tan⁻¹).

What is the hypotenuse?+

The longest side of a right-angled triangle, always opposite the right angle.

When do you use Pythagoras vs trigonometry?+

Use Pythagoras when you know two sides and want the third. Use trigonometry when angles are involved.

Related calculators

Sin Cos Tan Calculator

Calculate sin, cos and tan of any angle with working

Pythagoras Theorem Calculator

Find any side of a right triangle using a² + b² = c²

Angle Calculator

Find missing angles in polygons, triangles and on lines