Trig Identities Calculator
Enter an angle, pick degrees or radians, and this calculator instantly computes all six trigonometric functions (sin, cos, tan, csc, sec, cot) and verifies every major trig identity family for your angle - Pythagorean, quotient, reciprocal, double-angle, half-angle, sum and difference, cofunction, and negative-angle. Each identity is shown with the left side, right side, and step-by-step verification so you can follow the arithmetic at a glance.
What are trigonometric identities?
Trigonometric identities are equations involving trig functions that are true for every valid value of the angle. Unlike conditional equations, which hold only for specific angles, identities hold universally. They are used in calculus to simplify integrands, in physics to resolve wave equations, in engineering to analyse signals, and in geometry to prove relationships between sides and angles. The fundamental source of most identities is the unit circle: any point on a circle of radius 1 has coordinates (cos θ, sin θ), and the Pythagorean theorem applied to that point gives sin²θ + cos²θ = 1, the root from which most other identities grow.
The nine identity families and how to use them
Pythagorean identities (sin²θ + cos²θ = 1, and its two rearrangements for tan and sec) let you swap between sine and cosine. Reciprocal identities define csc, sec, and cot as the inverses of sin, cos, and tan. Quotient identities express tan and cot as ratios. Double-angle identities replace sin 2θ or cos 2θ with expressions in single-angle functions, which is essential in integration by parts. Half-angle identities work in the other direction, replacing a function of θ/2 with a function of θ, useful for power-reduction. Sum and difference identities expand sin(A+B) or cos(A+B) into products of sin and cos at separate angles - they underlie the Fourier transform. Cofunction identities swap sin for cos (and tan for cot, sec for csc) by replacing the angle with its complement. Negative-angle identities simplify expressions with negative inputs: sin is odd so sin(-θ) = -sin(θ), while cos is even so cos(-θ) = cos(θ). Triple-angle identities expand sin 3θ or cos 3θ into polynomials in the single-angle functions, used in Chebyshev-polynomial contexts.
Standard angles and their exact values
A small set of angles (0°, 30°, 45°, 60°, 90°, and their multiples) have exact trig values expressible without a calculator. At 30°, sin = 1/2, cos = sqrt(3)/2. At 45°, sin = cos = sqrt(2)/2. At 60°, sin = sqrt(3)/2, cos = 1/2. At 0° and 90° the values become 0 or 1. Memorising these benchmarks lets you verify identity derivations by hand and catch sign errors quickly. The calculator above shows you the numeric values for any angle you enter, which you can compare against known exact values to build intuition.
Tips for proving and verifying trig identities
The standard technique is to work on the more complex side of the equation and transform it step by step until it matches the simpler side - never manipulate both sides simultaneously, as that can introduce false proofs. Convert everything to sin and cos first if you are stuck, then factor or expand. Look for Pythagorean substitutions (replacing 1 - sin²θ with cos²θ, for example). Use the sum-to-product or product-to-sum formulas when you see sums of trig functions. This calculator numerically verifies any identity by evaluating both sides at your chosen angle, which is a fast sanity check even though a numerical match is not a formal proof.
Trigonometric identity families
| Family | Primary identity | Use case |
|---|---|---|
| Pythagorean | sin²θ + cos²θ = 1 | Simplify expressions, solve equations |
| Reciprocal | csc θ = 1/sin θ | Rewrite in terms of basic functions |
| Quotient | tan θ = sin θ / cos θ | Rewrite tangent and cotangent |
| Double-angle | sin 2θ = 2 sin θ cos θ | Expand double-frequency terms |
| Half-angle | sin(θ/2) = ±sqrt((1-cosθ)/2) | Integrate half-angle expressions |
| Sum and difference | sin(A+B) = sinA cosB + cosA sinB | Combine or split angles |
| Cofunction | sin θ = cos(90° - θ) | Convert sin/cos, tan/cot, sec/csc |
| Negative-angle | sin(-θ) = -sin θ | Evaluate negative angles |
| Triple-angle | sin 3θ = 3 sin θ - 4 sin³θ | Higher-order angle expansions |
The nine core identity families, with their primary formula and typical use case.
Frequently asked questions
What is the most important trig identity to know?
The Pythagorean identity sin²θ + cos²θ = 1 is the foundation of almost everything else. Dividing both sides by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives cot²θ + 1 = csc²θ. All three are essential for simplifying expressions and solving equations in calculus and physics.
How do I prove a trig identity step by step?
Pick the more complicated side, convert all functions to sin and cos, then simplify using algebra and the Pythagorean identity until you reach the other side. Avoid moving terms across the equals sign. This calculator shows the numeric values at both sides so you can confirm your algebraic steps are heading in the right direction.
What is the difference between degrees and radians?
Degrees divide a full circle into 360 equal parts. Radians measure arc length on a unit circle: a full circle is 2π radians (~6.2832). To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Calculus formulas and most scientific computing use radians, while geometry and navigation traditionally use degrees.
Why does tan(θ) become undefined at 90 degrees?
Tangent equals sin divided by cos. At 90°, cos(90°) = 0, so the division is undefined (no real number times zero equals sin 90° = 1). The same happens at 270° and at any odd multiple of 90°. On the unit circle these are the points where the tangent line is vertical, confirming there is no finite slope.
What are cofunction identities and when do I use them?
Cofunction identities pair complementary trig functions: sin θ = cos(90° - θ), tan θ = cot(90° - θ), and sec θ = csc(90° - θ). They are useful when you need to rewrite a function in terms of its complementary angle - common in triangle problems where two angles sum to 90°, and in simplifying expressions that mix sine and cosine of the same angle.
How do sum and difference identities work?
Sum and difference identities let you expand sin(A ± B) and cos(A ± B) into products: sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B - sin A sin B. They are the building blocks of the Fourier series (every sinusoid can be shifted using these identities) and appear frequently in integration and differential equations.