Right Triangle Trigonometry Calculator

Find all sides, angles, and trigonometric ratios of a right triangle using SOH-CAH-TOA.

Choose What You Know

Results

Triangle Solved
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all values found
sin(A)
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Opposite / Hypotenuse
cos(A)
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Adjacent / Hypotenuse
tan(A)
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Opposite / Adjacent
Side a (Opposite)--
Side b (Adjacent)--
Side c (Hypotenuse)--
Angle A--
Angle B--
Angle C (right)90°
Area--
Perimeter--

Step-by-Step Solution

What Is Right Triangle Trigonometry?

Right triangle trigonometry studies the relationships between the angles and sides of a right triangle (a triangle with one 90-degree angle). The three primary trigonometric ratios -- sine, cosine, and tangent -- relate each acute angle to the ratios of two specific sides.

These relationships are essential in fields such as engineering, physics, architecture, navigation, and computer graphics. Whenever you need to find an unknown distance or angle, right triangle trigonometry provides the tools.

SOH-CAH-TOA: The Core Mnemonic

SOH-CAH-TOA is the classic mnemonic for remembering the three basic trigonometric ratios:

SOH -- Sine

Sine of an angle equals the length of the Opposite side divided by the Hypotenuse.

sin(A) = Opposite / Hypotenuse

CAH -- Cosine

Cosine of an angle equals the length of the Adjacent side divided by the Hypotenuse.

cos(A) = Adjacent / Hypotenuse

TOA -- Tangent

Tangent of an angle equals the length of the Opposite side divided by the Adjacent side.

tan(A) = Opposite / Adjacent

The Pythagorean Theorem

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². This fundamental relationship lets you find the third side when you know two sides, even without knowing any angles.

Inverse Trigonometric Functions

When you know side lengths but not the angle, use inverse functions:

  • arcsin (sin&sup-1;): Given opposite and hypotenuse, find the angle.
  • arccos (cos&sup-1;): Given adjacent and hypotenuse, find the angle.
  • arctan (tan&sup-1;): Given opposite and adjacent, find the angle.

Common Right Triangles

30-60-90 Triangle

Side ratios are 1 : sqrt(3) : 2. The shortest side is opposite the 30-degree angle.

Sides: x, x*sqrt(3), 2x

45-45-90 Triangle

An isosceles right triangle with side ratios 1 : 1 : sqrt(2).

Sides: x, x, x*sqrt(2)

3-4-5 Triangle

The most common Pythagorean triple. Angles are approximately 36.87, 53.13, and 90 degrees.

3^2 + 4^2 = 5^2 (9+16=25)

Real-World Applications

  • Surveying: Calculating heights and distances using angle measurements.
  • Construction: Determining roof pitch, ramp angles, and structural loads.
  • Navigation: Finding bearings, distances, and positions.
  • Physics: Resolving force vectors into components.
  • Computer Graphics: Rotating objects, calculating shadows, and perspective projections.