Trig Triangle Calculator
Solve right triangles using trigonometry with sides, angles, or area
Solve Right Triangle
Triangle Solution
Trigonometric Ratios:
For Angle A (36.8699°):
sin(A) = a/c = 3/5 = 0.6
cos(A) = b/c = 4/5 = 0.8
tan(A) = a/b = 3/4 = 0.75
For Angle B (53.1301°):
sin(B) = b/c = 4/5 = 0.8
cos(B) = a/c = 3/5 = 0.6
tan(B) = b/a = 4/3 = 1.3333
Right Triangle
a: opposite to angle A
b: opposite to angle B
c: hypotenuse (longest side)
Key Formulas
Pythagorean Theorem:
c² = a² + b²
Trigonometric Ratios:
sin(A) = a/c
cos(A) = b/c
tan(A) = a/b
Area Formula:
Area = (a × b)/2
Quick Tips
In a right triangle, one angle is always 90°
The two acute angles always sum to 90°
The hypotenuse is always the longest side
Use SOHCAHTOA to remember trig ratios
Understanding Right Triangle Trigonometry
Three Ways to Solve
1. Two Sides Known
When you know two sides of a right triangle, you can find the third side using the Pythagorean theorem, then calculate angles using inverse trig functions.
- • Find third side: c = √(a² + b²)
- • Find angles: A = arctan(a/b)
- • Calculate area: Area = (a × b)/2
2. Angle and Side Known
With one angle and one side, use trigonometric ratios to find the other sides and angle.
- • Other angle: B = 90° - A
- • Use sin, cos, tan ratios
- • Calculate remaining sides
3. Area and Side Known
From the area formula and one known side, find the other leg, then solve as in method 1.
- • Area = (a × b)/2
- • Solve for unknown leg
- • Apply Pythagorean theorem
Trigonometric Relationships
SOHCAHTOA
SOH: sin(θ) = Opposite/Hypotenuse
CAH: cos(θ) = Adjacent/Hypotenuse
TOA: tan(θ) = Opposite/Adjacent
Inverse Functions
θ = arcsin(Opposite/Hypotenuse)
θ = arccos(Adjacent/Hypotenuse)
θ = arctan(Opposite/Adjacent)
Special Right Triangles
45-45-90: sides in ratio 1:1:√2
30-60-90: sides in ratio 1:√3:2
Real-World Applications
Construction & Engineering
Calculate roof angles, ramp slopes, structural supports, and building heights using trigonometric principles.
Navigation & Surveying
Determine distances, elevations, and directions in GPS systems, land surveying, and marine navigation.
Physics & Astronomy
Analyze force vectors, projectile motion, wave properties, and celestial object positions.