Sum and Difference Identities Calculator
Welcome to our sum and difference identities calculator, where we study the sum and difference formulas for all six trigonometric functions — sine, cosine, tangent, cotangent, secant, and cosecant.
Enter two angles (α and β) in degrees (American system) or radians (metric system), and instantly get all twelve identities: six for sums (α + β) and six for differences (α − β).
Trigonometric Sum and Difference Formulas
Sum and difference identities express a trig function of a combined angle through functions of the individual angles:
Sine
sin(α + β) = sin α · cos β + cos α · sin β
sin(α − β) = sin α · cos β − cos α · sin β
Cosine
cos(α + β) = cos α · cos β − sin α · sin β
cos(α − β) = cos α · cos β + sin α · sin β
Tangent
tan(α + β) = (tan α + tan β) / (1 − tan α · tan β)
tan(α − β) = (tan α − tan β) / (1 + tan α · tan β)
Undefined when the denominator equals zero, i.e., tan α · tan β = 1.
Cotangent
cot(α + β) = (cot α · cot β − 1) / (cot β + cot α)
cot(α − β) = (cot α · cot β + 1) / (cot β − cot α)
Secant and Cosecant
sec(α ± β) = 1 / cos(α ± β)
csc(α ± β) = 1 / sin(α ± β)
Angle Units: Degrees vs. Radians
The calculator supports two systems:
- Degrees (°) — the American / customary system. A full circle = 360°. Common angles: 30°, 45°, 60°, 90°.
- Radians (rad) — the metric / SI system used in science and mathematics. A full circle = 2π ≈ 6.2832 rad. Common angles: π/6, π/4, π/3, π/2.
To convert: degrees × π / 180 = radians, and radians × 180 / π = degrees.
Special Angle Reference Table
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undefined |
Example: sin(75°) using the Sum Formula
Express 75° = 45° + 30°:
sin(75°) = sin(45° + 30°)
= sin 45° · cos 30° + cos 45° · sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2) / 4 ≈ 0.9659
Example: cos(15°) using the Difference Formula
Express 15° = 45° − 30°:
cos(15°) = cos(45° − 30°)
= cos 45° · cos 30° + sin 45° · sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2) / 4 ≈ 0.9659
Double Angle Formulas (Special Case)
When α = β, the sum formulas yield the double angle formulas:
sin(2α) = 2 sin α · cos α
cos(2α) = cos²α − sin²α = 1 − 2sin²α = 2cos²α − 1
tan(2α) = 2 tan α / (1 − tan²α)
Frequently Asked Questions
What are sum and difference identities used for?
They let you compute the exact value of a trig function at a "difficult" angle by expressing it as the sum or difference of two "easy" angles (multiples of 30°, 45°, 60°, or 90°).
Why does the calculator show "Undefined" for some values?
Certain combinations make the denominator of tan, cot, sec, or csc equal to zero. For example, tan(90°) is undefined because cos(90°) = 0.
Can I enter negative angles?
Yes. Negative angles rotate clockwise. For example, sin(−30°) = −sin(30°) = −0.5.
Is this calculator suitable for studying trigonometry?
Absolutely. The calculator shows both numerical results and the formulas, making it ideal for students, teachers, and engineers working with oscillations, waves, or coordinate geometry.