📐Trigonometry Calculator
Compute all six trigonometric functions (sin, cos, tan, csc, sec, cot) for any angle in degrees or radians. Or use the right triangle solver: enter any two known values (sides a, b, c, angles α, β, or area) and instantly solve for all the rest.
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Trig Function Values (mode: trig functions) / Triangle Sides (mode: right triangle)
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Trigonometry Calculator: Six Trig Functions and Right Triangle Solver
The six trigonometric functions relate angles to ratios of sides in a right triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj, and their reciprocals csc, sec, cot. For a right triangle with legs a, b and hypotenuse c: the Pythagorean theorem gives a² + b² = c², the angles sum to α + β = 90°, and area = ½ab.
SOH-CAH-TOA: Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
This calculator has two modes: Trig Functions computes all six trigonometric values for any angle in degrees or radians; Right Triangle Solver finds all unknown sides, angles, and area given any two independent known values.
The Six Trigonometric Functions
Given a right triangle with angle θ: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent = sin θ/cos θ; csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ = cos θ/sin θ. Key identities: sin²θ + cos²θ = 1 (Pythagorean identity); tan(90°−θ) = cot θ (co-function); sin(−θ) = −sin θ (odd function); cos(−θ) = cos θ (even function).
Solving Right Triangles
Any two independent pieces of information determine a unique right triangle (if at least one is a side length). The solver handles all valid combinations: two sides (use Pythagorean theorem for third, arctan for angles); a side and an angle (use sin/cos/tan to find other sides); area and a side (find other leg, then use Pythagorean theorem); and more. The algorithm runs iterative constraint propagation — applying every applicable formula repeatedly until all unknowns are determined.
Frequently Asked Questions
What does SOH-CAH-TOA mean?
SOH-CAH-TOA is a memory aid for the three main trig functions: SOH = Sin(θ) = Opposite/Hypotenuse; CAH = Cos(θ) = Adjacent/Hypotenuse; TOA = Tan(θ) = Opposite/Adjacent. "Opposite" is the side opposite the angle, "Adjacent" is the side next to the angle (not the hypotenuse), and "Hypotenuse" is the longest side (opposite the right angle). Example: in a 3-4-5 triangle with angle α opposite the side of length 3: sin α = 3/5 = 0.6; cos α = 4/5 = 0.8; tan α = 3/4 = 0.75; α = arcsin(0.6) ≈ 36.87°.
How do I solve a right triangle?
To solve a right triangle (find all sides and angles), you need any 2 of the 6 values (sides a, b, c or angles α, β), with at least one being a side. Common cases: (1) Two sides given: use a²+b²=c² for the third side; use arctan(a/b) for angles. (2) Hypotenuse and one angle: side = c×sin(α) or c×cos(α). (3) One leg and one angle: other leg = a×tan(α) or a/tan(α); hypotenuse = a/sin(α). (4) Two legs: c=√(a²+b²), α=arctan(a/b), β=90°−α. Always remember: the two acute angles must sum to 90°.
What are the exact values of sin, cos, and tan for 30°, 45°, and 60°?
These are the "special angles" with exact values: sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3=√3/3. sin 45°=cos 45°=√2/2, tan 45°=1. sin 60°=√3/2, cos 60°=1/2, tan 60°=√3. These come from the 30-60-90 triangle (sides 1, √3, 2) and the 45-45-90 triangle (sides 1, 1, √2). At 0°: sin=0, cos=1, tan=0. At 90°: sin=1, cos=0, tan=undefined. At 180°: sin=0, cos=−1, tan=0.
What is the difference between degrees and radians?
Degrees divide a full circle into 360 equal parts. Radians measure angles in terms of arc length: one radian is the angle where the arc length equals the radius. A full circle = 2π radians = 360°. Conversion: radians = degrees × π/180; degrees = radians × 180/π. Key conversions: 0°=0, 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π, 270°=3π/2, 360°=2π. Radians are preferred in calculus because formulas are simpler: d/dθ(sin θ) = cos θ only works with radians. Most calculators default to degrees for practical geometry work.