Trigonometry Calculator
Calculate trigonometric functions, inverse functions, and solve triangles
Trigonometric Functions
Understanding Trigonometry
Trigonometry is the study of relationships between angles and sides of triangles. The word comes from Greek: 'trigonon' (triangle) and 'metron' (measure). It's fundamental to mathematics, physics, engineering, astronomy, and many other fields.
The six trigonometric functions - sine, cosine, tangent, cosecant, secant, and cotangent - relate an angle to ratios of sides in a right triangle. These functions extend beyond triangles to describe periodic phenomena like waves, oscillations, and circular motion.
Trigonometry connects geometry to algebra and enables solutions to problems involving angles, distances, and periodic behavior. From calculating building heights to modeling sound waves, trigonometry is everywhere.
The Six Trigonometric Functions
Sine (sin)
Opposite / Hypotenuse. Primary function for waves and oscillations.
Cosine (cos)
Adjacent / Hypotenuse. 90° phase shift from sine.
Tangent (tan)
Opposite / Adjacent = sin/cos. Slope of a line at angle θ.
Reciprocals
csc = 1/sin, sec = 1/cos, cot = 1/tan.
Common Angle Values
Memorize these special angle values for quick reference:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° (0) | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | 1/√3 |
| 45° (π/4) | √2/2 | √2/2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | undefined |
| 180° (π) | 0 | -1 | 0 |
Key Trigonometric Identities
Pythagorean Identities
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. Derived from the Pythagorean theorem.
Sum/Difference Formulas
sin(A±B) = sinA·cosB ± cosA·sinB. cos(A±B) = cosA·cosB ∓ sinA·sinB. Essential for combining angles.
Double Angle Formulas
sin(2θ) = 2sinθ·cosθ. cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.
Co-function Identities
sin(90°-θ) = cosθ, cos(90°-θ) = sinθ. Functions of complementary angles are related.
Frequently Asked Questions
When should I use degrees vs radians?
Use degrees for everyday angles (architecture, navigation). Use radians for calculus, physics, and advanced math - they simplify formulas. π radians = 180°.
Why is tan(90°) undefined?
tan(90°) = sin(90°)/cos(90°) = 1/0. Division by zero is undefined. The tangent function has vertical asymptotes at 90°, 270°, etc.
What are inverse trig functions?
arcsin, arccos, arctan (also written sin⁻¹, cos⁻¹, tan⁻¹) find the angle given a ratio. If sin(30°) = 0.5, then arcsin(0.5) = 30°.
How do I solve a triangle?
Use Law of Sines (a/sinA = b/sinB = c/sinC) when you know an angle and its opposite side. Use Law of Cosines (c² = a² + b² - 2ab·cosC) for other cases.
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