Trigonometric Identities Calculator
Verify and compute trigonometric identities including sine/cosine sum formulas and double-angle formulas. Use it when studying trig, checking homework, or confirming angle transformations.
About this calculator
Trigonometric identities are equations that hold true for all valid angle values. This calculator covers four key identities. The sine sum formula states: sin(A+B) = sin A·cos B + cos A·sin B. The cosine sum formula states: cos(A+B) = cos A·cos B − sin A·sin B. The double-angle sine formula is: sin(2A) = 2·sin A·cos A. The double-angle cosine formula is: cos(2A) = cos(2A), evaluated directly. These identities are derived from the unit circle and are foundational in algebra, calculus, and physics. Selecting the identity type and entering your angles lets the calculator return the exact numerical result of the right-hand side expression.
How to use
Suppose you want to verify the sine sum identity for A = 30° and B = 45°. Select 'Sum – Sine' as identity type and enter angle1 = 30, angle2 = 45 in degrees. The calculator evaluates: sin(30°)·cos(45°) + cos(30°)·sin(45°) = 0.5·0.7071 + 0.8660·0.7071 = 0.3536 + 0.6124 ≈ 0.9659. You can verify this equals sin(75°) ≈ 0.9659. Try the double-angle sine with angle1 = 30°: 2·sin(30°)·cos(30°) = 2·0.5·0.8660 ≈ 0.8660, which equals sin(60°).
Frequently asked questions
What is the sine sum formula and how is it derived?
The sine sum formula states sin(A+B) = sin A·cos B + cos A·sin B. It is derived geometrically using the unit circle by projecting a rotated vector onto the sine axis. This identity is fundamental because it lets you express the sine of a combined angle using individual angle values. It also underpins the derivation of double-angle and half-angle formulas.
How does the double-angle cosine formula differ from the sum formula?
The double-angle cosine formula cos(2A) is a special case of the cosine sum formula where both angles equal A: cos(A+A) = cos²A − sin²A. Using the Pythagorean identity, this can also be written as 2cos²A − 1 or 1 − 2sin²A. All three forms are equivalent, and the choice depends on which variables are known in your problem. This calculator evaluates cos(2A) directly for clarity.
When should I use trigonometric identities instead of a regular trig calculator?
Use identities when you need to simplify an expression, prove an equation, or compute a trig value for a non-standard angle by breaking it into known parts. For example, sin(75°) is not on a standard table, but you can compute it as sin(45°+30°) using the sum formula. Identities are also essential when solving integrals or differential equations that contain trig functions in complex forms.