Trig Identity Solver
This tool takes any angle, in degrees or radians, and instantly works out every standard trigonometric identity for that angle so you can check homework, verify a derivation, or explore how the identities fit together without doing the algebra by hand. You type in an angle value and choose degrees or radians, and everything below updates live. First you get the angle converted into both units, then the six trigonometric functions: sine, cosine, tangent, cosecant, secant and cotangent, calculated to six decimal places. Next come the three Pythagorean identities, each shown with its left-hand-side value and a tick or cross confirming whether it truly equals the right-hand side for your angle. The solver then evaluates the double angle formulas for sin(2θ), cos(2θ) and tan(2θ), and the half angle formulas for sin(θ/2), cos(θ/2) and tan(θ/2), automatically working out the correct plus or minus sign from the quadrant your half angle falls in. A reciprocal and quotient identities panel spells out how tangent, cotangent, secant and cosecant relate back to sine and cosine, and a worked example at 30° shows every calculation in full. Where a function is undefined at your chosen angle, such as tangent at 90°, the solver clearly labels the result as undefined rather than showing an error.
1. Your Angle
2. Angle in Both Units
Pythagorean Identities
| Identity | Formula | LHS value | = RHS? |
|---|---|---|---|
| First | sin²(θ) + cos²(θ) = 1 | 1.000000 | ✓ |
| Second | 1 + tan²(θ) = sec²(θ) | 1.333333 | ✓ |
| Third | 1 + cot²(θ) = csc²(θ) | 4.000000 | ✓ |
Double Angle Formulas (for 2θ)
| Function | Formula | Value at 2θ |
|---|---|---|
| sin(2θ) | 2 sin(θ) cos(θ) | 0.866025 |
| cos(2θ) | cos²(θ) − sin²(θ) | 0.500000 |
| tan(2θ) | 2 tan(θ) / (1 − tan²(θ)) | 1.732051 |
Half Angle Formulas (for θ/2)
| Function | Formula | Value at θ/2 | Note |
|---|---|---|---|
| sin(θ/2) | ±√((1 − cos θ) / 2) | 0.258819 | + (Q1 for θ/2) |
| cos(θ/2) | ±√((1 + cos θ) / 2) | 0.965926 | + (Q1 for θ/2) |
| tan(θ/2) | sin θ / (1 + cos θ) | 0.267949 |
Reciprocal and Quotient Identities
Reciprocal
Quotient
Worked Example: θ = 30°
sin(30°) = 0.500000; cos(30°) = 0.866025; tan(30°) = 0.577350
csc(30°) = 2.000000; sec(30°) = 1.154701; cot(30°) = 1.732051
Pythagorean: sin²(30°) + cos²(30°) = 0.25 + 0.75 = 1.000000
Double angle: sin(60°) = 2 × 0.5 × 0.866025 = 0.866025; cos(60°) = 0.75 − 0.25 = 0.500000
Half angle: sin(15°) = √((1 − 0.866025) / 2) = √(0.066987) = 0.258819
About Trigonometric Identities
A trigonometric identity is an equation involving trig functions that holds true for every valid value of the variable. Unlike equations that you solve for a specific value, identities are true everywhere they are defined. They are used to simplify expressions, prove other results, solve trig equations, and evaluate integrals in calculus.
The Six Trigonometric Functions
For an angle θ on the unit circle (radius = 1), the six trig functions are defined as follows:
| Function | Symbol | Definition (unit circle) | Reciprocal of |
|---|---|---|---|
| Sine | sin(θ) | y-coordinate of point on unit circle | csc |
| Cosine | cos(θ) | x-coordinate of point on unit circle | sec |
| Tangent | tan(θ) | sin(θ) / cos(θ) | cot |
| Cosecant | csc(θ) | 1 / sin(θ) | sin |
| Secant | sec(θ) | 1 / cos(θ) | cos |
| Cotangent | cot(θ) | cos(θ) / sin(θ) | tan |
Pythagorean Identities
The three Pythagorean identities follow directly from the Pythagorean theorem applied to a point on the unit circle. Because x² + y² = 1 for any point on a unit circle, and x = cos(θ), y = sin(θ), we get sin²(θ) + cos²(θ) = 1. Dividing through by cos²(θ) gives 1 + tan²(θ) = sec²(θ). Dividing through by sin²(θ) gives cot²(θ) + 1 = csc²(θ). These identities are used constantly in simplifying trig expressions and solving equations.
Double Angle Formulas
The double angle formulas are derived from the angle addition identities by setting both angles equal. From sin(A + B) = sin A cos B + cos A sin B, putting A = B = θ gives sin(2θ) = 2 sin(θ) cos(θ). From cos(A + B) = cos A cos B − sin A sin B, putting A = B = θ gives cos(2θ) = cos²(θ) − sin²(θ), which can also be written as 2cos²(θ) − 1 or 1 − 2sin²(θ). The tangent double angle formula is tan(2θ) = 2tan(θ) / (1 − tan²(θ)).
Half Angle Formulas
The half angle formulas are derived from the double angle cosine identities. Starting from cos(2θ) = 1 − 2sin²(θ), replacing θ with θ/2 and rearranging gives sin(θ/2) = ±√((1 − cos θ) / 2). Similarly, cos(θ/2) = ±√((1 + cos θ) / 2). The sign depends on which quadrant θ/2 falls in. The tangent half angle formula can be expressed as tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ, which avoids the sign ambiguity.
Degrees vs Radians
Angles can be measured in degrees (one full revolution = 360°) or radians (one full revolution = 2π radians). To convert from degrees to radians, multiply by π/180. To convert from radians to degrees, multiply by 180/π. Radians are the natural unit for calculus because the derivative of sin(x) is cos(x) only when x is in radians.
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Sources and method: Trigonometric identities sourced from standard mathematics references: Stewart, J. (2016) Calculus: Early Transcendentals, 8th ed., Appendix D. All values computed in JavaScript using native Math.sin, Math.cos, and Math.tan (IEEE 754 double precision), with angles converted to radians via θ × π / 180. Half angle signs are determined by the quadrant of θ/2.
This tool evaluates numerical values of trig identities for a given angle. It does not perform symbolic algebraic simplification or prove identities for general expressions. Values of tan, csc, sec, and cot are undefined at certain angles (for example, tan at 90°); the solver shows "undefined" at those points.