Trig Identities Calculator - Pythagorean Sum Difference Cofunction Solver
Use this trig identities calculator to enter alpha and beta in degrees or radians and read the six sum and difference identities plus the Pythagorean check.
Trig Identities Calculator
Results
What Is Trig Identities Calculator?
A trig identities calculator is a trigonometry tool that takes two input angles, alpha and beta, and returns every core trigonometric identity at those inputs in one place. The page prints the Pythagorean identity check on alpha and beta, the six sum and difference identities for sin, cos, and tan, and a by-formula verification row beside each direct sine and cosine value.
- • Verifying a hand derivation: Plug in alpha and beta, then read the direct trig value next to the by-formula row to confirm the identity holds.
- • Comparing sum and difference results: Use the (alpha + beta) and (alpha - beta) blocks to see how the sign of beta changes the answer.
- • Teaching the Pythagorean identity: Show sin squared plus cos squared stays equal to 1 for both alpha and beta.
- • Recovering double angle results: Set beta equal to alpha to recover sin(2*alpha) and cos(2*alpha) from the sum block.
The same page covers the identities students usually look up separately: Pythagorean, angle addition, and angle subtraction.
A trig identities calculator like this one keeps the formula box and the verification rows side by side, so the results panel shows whether the identity holds.
If you want to focus only on the six sum and difference identities and skip the Pythagorean check, the Sum Difference Identities Calculator returns the same six outputs.
How Trig Identities Calculator Works
The solver reads alpha, beta, and the unit selector, converts both inputs to radians, then evaluates the Pythagorean identity, the six sum and difference identities, and the by-formula verification rows for sin and cos.
- alpha (α): First input angle, in degrees or radians based on the unit selector.
- beta (β): Second input angle, in degrees or radians based on the unit selector.
- Composite angle (α ± β): The angle used by the sum and difference identity rows.
- Pythagorean identity: sin squared plus cos squared equals 1, evaluated on alpha and beta as a parse and algebra check.
The by-formula row on each sine and cosine output is computed independently from the direct trig value, so any rounding gap is visible. Tangent has no such row because the identity has a removable denominator.
The Pythagorean identity is checked on both inputs because sin squared plus cos squared equals 1 for every real number, so a value noticeably different from 1 means the input failed to parse rather than the unit selector being wrong.
Worked example: α = 30°, β = 45°
α = 30, β = 45, unit = degrees
α + β = 75° and α − β = −15°. sin(75°) ≈ 0.9659 and cos(75°) ≈ 0.2588 by the addition identities. sin(−15°) ≈ −0.2588 and cos(−15°) ≈ 0.9659 by the subtraction identities.
sin(α + β) ≈ 0.9659; cos(α + β) ≈ 0.2588; tan(α + β) ≈ 3.7321; sin(α − β) ≈ −0.2588; cos(α − β) ≈ 0.9659; tan(α − β) ≈ −0.2679; sin²(α) + cos²(α) = 1; sin²(β) + cos²(β) = 1.
The by-formula rows match the direct rows to four decimals, both Pythagorean rows equal 1, and re-entering α = 0.7854 and β = 0.5236 in radians reproduces the same numbers since both inputs land on the same interior angles.
According to Wolfram MathWorld: Trigonometric Identities, sin(α + β) = sin(α)cos(β) + cos(α)sin(β), cos(α + β) = cos(α)cos(β) − sin(α)sin(β), and tan(α + β) = (tan(α) + tan(β)) / (1 − tan(α)tan(β)) all follow from the addition formulas.
Set β = α on this page to recover the double angle block, then cross-check the same numbers against the Double Angle Identities Calculator which expands sin(2θ), cos(2θ), and tan(2θ).
Key Concepts Explained
Four concepts make the rest of the page easy to follow. They are the same building blocks the identities are derived from.
Pythagorean identity
sin²θ + cos²θ = 1 holds for every real θ, so the calculator prints it for alpha and beta as a parse and algebra check. The result is 1 in radians regardless of the input unit, so a value different from 1 points to a non-numeric field.
Angle addition identities
Each sum identity rewrites sin, cos, and tan of a composite angle as a sum of products of the trig values of the two input angles.
Euler's formula derivation
Multiplying e to the i alpha and e to the i beta gives e to the i (alpha + beta), and matching the real and imaginary parts produces all four sine and cosine identities in two lines.
Symmetry of sine and cosine
Sine is an odd function, so sin(-x) = -sin(x). Cosine is an even function, so cos(-x) = cos(x). That pair of facts flips the sign of the second term in the difference identities.
These four ideas explain why every identity on the page is connected: the Pythagorean identity powers the cosine forms, the angle addition identities power the sum and difference rows, and Euler's formula produces all four sine and cosine identities at once.
When α + β equals 90°, the difference identity collapses to cos(β) = sin(α) and sin(β) = cos(α), the same pattern the Cofunction Calculator uses to swap sine and cosine.
How to Use This Calculator
Working with the calculator takes a few seconds. Pick the unit, type both angles, and read the full identity table plus the by-formula rows.
- 1 Pick the unit: Choose degrees for 30, 45, 60, or 90. Choose radians for the decimal equivalent of the same angle, such as 0.5236, 0.7854, 1.0472, or 1.5708. The fields are numeric only, so write π/4 as 0.7854.
- 2 Enter alpha: Type the first angle. The default 30° produces sin(75°) and cos(75°) from beta = 45°.
- 3 Enter beta: Type the second angle. Beta is added to and subtracted from alpha inside the identity rows.
- 4 Read the Pythagorean block: The top of the panel shows sin²(α) + cos²(α) and sin²(β) + cos²(β). Both equal 1 for any finite numeric angle pair, since the identity simplifies to 1 in radians.
- 5 Read the (α + β) block: The sin(α + β), cos(α + β), and tan(α + β) rows show the direct trig value. By-formula verification rows sit beside the sine and cosine values only; tangent has no such row because the identity has a removable denominator.
- 6 Read the (α − β) block: The sin(α − β), cos(α − β), and tan(α − β) rows follow the same pattern. Sine flips sign and cosine keeps sign, the signature of sin(-x) = -sin(x) and cos(-x) = cos(x).
Suppose a problem gives α = 60° and β = 30° and asks for sin(α + β) plus the difference values. Enter 60 and 30 with degrees selected, then read sin(90°) = 1, cos(90°) = 0, tan(90°) = 'undefined', and the difference block showing sin(30°) = 0.5 and cos(30°) ≈ 0.8660.
If you only need a single sine or tangent value at one angle, the Trig Function Calculator returns the same numeric value without the identity table.
Benefits of Using This Calculator
A single screen that lists every trig identity most students look up separately makes this trig identities calculator useful for homework.
- • All major identities in one view: The Pythagorean check, the six sum identities, and the six difference identities live on the same panel.
- • Direct value next to by-formula row: Each sine and cosine identity has a right-hand-side row beside the direct value, which makes the formula visible and confirms the derivation at a glance.
- • Built-in Pythagorean check: sin²(α) + cos²(α) and sin²(β) + cos²(β) run at the top. A non-numeric input such as a stray letter shows up as a value different from 1.
- • Sign pattern visible across blocks: Sine flips sign and cosine keeps sign when beta becomes minus beta. The two side-by-side blocks turn that symmetry into a visual check.
- • Tangent undefined handled cleanly: tan(α + β) and tan(α − β) report 'undefined' when the denominator is essentially zero. Tangent has no separate by-formula row for the same reason.
The biggest practical win is keeping the Pythagorean check, direct values, and by-formula rows on the same screen.
Setting β to half of α produces the half-angle identity values from the difference block, which is the same step the Half Angle Calculator runs on a single input.
Factors That Affect Your Results
Most of the time the trig identities calculator returns the expected values, but a few factors control what the tool can and cannot do.
Unit selector drives every conversion
Pick degrees for 30, 45, 60, or 90. Pick radians for the decimal equivalent, such as 0.5236, 0.7854, 1.0472, or 1.5708. The wrong unit choice silently produces the wrong composite angle.
Tangent denominator vanishes at 90° + k·180°
tan(α + β) divides sine by cosine, so any composite angle on 90°, 270°, or 450° makes the tangent row report 'undefined'.
Floating-point rounding near the boundary
Inputs like 59.999999° can land tan(α + β) just inside the undefined region. The solver treats |cos(α + β)| below 1e-12 as undefined.
Sign pattern between (α + β) and (α − β)
Sine flips sign when beta becomes minus beta, cosine keeps sign, and tangent flips the cross term sign.
- • The tool returns the principal real values of the six sum and difference identities plus the Pythagorean identity check. It does not compute complex-valued trig outputs.
- • Floating-point arithmetic means the by-formula rows match the direct rows only to roughly 15 significant digits.
If a problem demands an exact symbolic answer, work the identities by hand and use the calculator to confirm the numeric value.
According to Wikipedia: List of trigonometric identities, sin²θ + cos²θ = 1 holds for every real θ and turns cos²θ − sin²θ into 1 − 2 sin²θ and 2 cos²θ − 1.
According to Khan Academy: Trig identities, sin(π/2 − θ) = cos(θ), cos(π/2 − θ) = sin(θ), and tan(π/2 − θ) = cot(θ) all follow from the difference identities with α = π/2 and β = θ.
The power reduction identities are the algebraic reverse of the cosine sum and difference identities, and the Power Reducing Calculator prints those rows from a single angle input.
Frequently Asked Questions
Q: What is a trig identities calculator?
A: A trig identities calculator is a trigonometry tool that takes two input angles, alpha and beta, and returns the major trigonometric identities evaluated at those inputs. A complete version prints the Pythagorean identity check plus the six sum and difference identities for sin, cos, and tan.
Q: What are the main trigonometric identities?
A: The main trigonometric identities are the Pythagorean identity sin squared plus cos squared equals 1, the angle addition identities for sin, cos, and tan, the angle subtraction identities, and the cofunction identities. The page prints a by-formula verification row next to each direct sine and cosine value, since the tangent identity has a removable denominator and is not shown as a separate row.
Q: How do you verify the Pythagorean identity with two angles?
A: Compute sin squared plus cos squared for alpha and sin squared plus cos squared for beta. Both rows should equal 1 for any real angle pair, because sin squared plus cos squared simplifies to 1 in radians regardless of the original input unit. A value far from 1 means the field is not numeric, not that the unit selector is wrong.
Q: Why does tan(α + β) sometimes become undefined?
A: tan(α + β) = (tan(α) + tan(β)) / (1 − tan(α)tan(β)) is undefined whenever the denominator is zero, which is the same as cos(α + β) being zero. The cleanest example is α + β = 90 degrees, where the cosine output is 0 and the tangent row is reported as 'undefined'.
Q: What is the difference between sum and difference identities?
A: The sum identities give the trig values of (α + β) and the difference identities give the trig values of (α − β). The difference identities are derived from the sum identities by replacing β with −β and using sin(−x) = −sin(x) and cos(−x) = cos(x).
Q: How do you derive trig identities from Euler's formula?
A: Write e to the i α = cos(α) + i sin(α) and e to the i β = cos(β) + i sin(β). Multiply to get e to the i (α + β), then match the real and imaginary parts of the expansion. That recovers all four sine and cosine sum identities in two lines.