What is a trigonometric identity?

A trig identity is an equation involving trig functions that holds true for every value of the variable. Unlike a conditional equation (which is only satisfied by specific angles), an identity is universally valid — for example sin²θ + cos²θ = 1 is always 1 regardless of θ. This calculator evaluates the numeric value of each identity at the angle you supply, letting you see the equality in action and check your own working.

What are the three Pythagorean identities?

The three Pythagorean identities follow directly from the unit circle. sin²θ + cos²θ = 1 is the fundamental one. Dividing through by cos²θ gives 1 + tan²θ = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ. This calculator evaluates all three (and their rearrangements) simultaneously so you can see how they relate numerically at any angle.

How does the double-angle formula work?

Double-angle formulas express a trig function of 2θ in terms of functions of θ alone. The most common are sin(2θ) = 2 sinθ cosθ and the three forms of cos(2θ): cos²θ − sin²θ, 1 − 2sin²θ, and 2cos²θ − 1. The calculator shows all forms side by side, confirming they are identical. These appear frequently in integration, solving equations, and Fourier analysis.

When do I use sum-and-difference vs product-to-sum formulas?

Sum-and-difference formulas — sin(A±B), cos(A±B), tan(A±B) — let you expand or combine trig functions when you know two separate angles. Product-to-sum formulas convert a product like sinA cosB into a sum or difference of single-argument functions: 2sinA cosB = sin(A+B) + sin(A−B). Product-to-sum is especially useful in calculus when integrating products of trig functions, and in signal processing where beating/interference patterns arise.

What is the half-angle formula and when is it useful?

Half-angle formulas give sin(θ/2), cos(θ/2), and tan(θ/2) in terms of cosθ. For example sin(θ/2) = ±√((1−cosθ)/2) and cos(θ/2) = ±√((1+cosθ)/2). The ± sign depends on the quadrant of θ/2. These are used to find exact values of angles like 15°, 22.5°, and 75°, and to simplify integrals of the form ∫ dθ / (a + b cosθ) via the Weierstrass t-substitution t = tan(θ/2).

What does "Solve for Angle" do?

Instead of evaluating a function at a known angle, "Solve for Angle" works in reverse. You supply a value x and choose which function equals x, and the calculator returns the principal value of θ using the relevant inverse function: arcsin for sin/csc, arccos for cos/sec, arctan for tan/cot. Remember that inverse trig functions return only the principal value — arcsin returns angles in [−90°, 90°], arccos in [0°, 180°], and arctan in (−90°, 90°). Add multiples of 360° (or 2π) to find the full solution set.

Can this calculator handle angles larger than 360° or negative angles?

Yes. JavaScript Math functions accept any real angle, and trig functions are periodic. sin(405°) = sin(45°) = 0.7071, for example, and the calculator returns the correct value automatically. If you are using the Solve panel, the principal-value output will be within the standard range of the inverse function, but the working identity remains valid for all real inputs.

Trigonometric Identity Calculator

Trigonometry is built on a rich web of equalities — the trigonometric identities — that hold for every angle without exception. Where a conditional equation like sin(θ) = 0.5 has only specific solutions, an identity such as sin²θ + cos²θ = 1 is valid for every real number θ. This calculator brings all the major identity families into one place and computes them numerically so you can verify your working, spot patterns, and build intuition.

What this calculator covers

Nine identity families are available from the dropdown:

Basic trig functions — sin, cos, tan, csc, sec, cot at your chosen angle.
Pythagorean identities — sin²θ + cos²θ = 1 and its sec/csc variants, all confirmed simultaneously.
Double-angle formulas — all four forms of sin(2θ) and cos(2θ), plus tan(2θ).
Half-angle formulas — sin(θ/2), cos(θ/2), tan(θ/2) in both the radical form and the quotient form.
Sum and difference — sin(A±B), cos(A±B), tan(A±B) using two separate angles A and β.
Product-to-sum — converts 2sinA cosB, 2cosA cosB, 2sinA sinB into sums of single-argument functions.
Power-reduction — expresses sin²θ, cos²θ, sin³θ, cos³θ, sin⁴θ, cos⁴θ in terms of multiples of θ.
Co-function identities — sin(θ) = cos(90°−θ) and the odd/even symmetry rules.
Inverse trig — evaluates arcsin(sin θ), arccos(cos θ), arctan(tan θ) and related compositions.
Solve for angle — given a value x and a function name, returns the principal angle θ such that fn(θ) = x.

How it works

All computation runs in your browser using JavaScript’s built-in Math.sin, Math.cos, Math.tan, Math.asin, Math.acos, and Math.atan — all operating in radians. Degree inputs are multiplied by π/180 before use. Results are rounded to 8 significant figures using parseFloat(n.toFixed(8)), which strips trailing zeros and shows only meaningful precision.

For the product-to-sum panel the calculator shows both the left-hand side (e.g. 2 sinA cosB) and the right-hand side (sin(A+B) + sin(A−B)) so you can confirm they match numerically. Any result that is mathematically undefined (e.g. tan(90°)) is displayed as “undefined” rather than Infinity.

Worked example — double-angle identities at θ = 30°

Set the unit to Degrees, select Double-Angle, and enter 30. You should see:

Identity	Value
sin(2θ) = 2sinθ cosθ	0.8660254
cos(2θ) = cos²−sin²	0.5
cos(2θ) = 1−2sin²	0.5
cos(2θ) = 2cos²−1	0.5
tan(2θ) = 2tan/(1−tan²)	1.7320508

All three forms of cos(60°) return exactly 0.5, confirming the identities. The value 1.7320508 for tan(60°) is √3, which you can verify: √3 ≈ 1.7320508.

Worked example — solving for an angle

Switch to Solve for Angle, choose sin, and enter 0.5. The result is 30° (π/6 radians) — the principal value. If the problem domain requires all solutions, you add 360°·n (for sine: also 150° + 360°·n) by hand.

Formula note

The identities computed here follow the SOHCAHTOA definitions of sine, cosine, and tangent on the unit circle. Angles are unrestricted real numbers; co-terminal angles (differing by multiples of 360°) yield identical trig-function values. The inverse functions return principal values only: arcsin ∈ [−90°, 90°], arccos ∈ [0°, 180°], arctan ∈ (−90°, 90°). This is standard for all calculators and programming languages.