Math
Sum and Difference Identities Calculator
The sum and difference identities expand sin, cos, and tan of α + β and α − β in terms of the functions of α and β.
Enter two angles to see both expansions with exact radicals, or enter a single angle like 75° or 15° and the calculator chooses building blocks such as 45° + 30° to derive its exact value.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
sin(α + β)
—
- cos(α + β)
- —
- tan(α + β)
- —
- sin(α − β)
- —
- cos(α − β)
- —
- tan(α − β)
- —
- α + β
- —
More details (1 more)
- α − β
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
The identities
Exact values of 15°, 75°, 105°, and friends
| Angle | Built as | sin | cos | tan |
|---|---|---|---|---|
| 15° | 45° − 30° | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 |
| 75° | 45° + 30° | (√6 + √2)/4 | (√6 − √2)/4 | 2 + √3 |
| 105° | 60° + 45° | (√6 + √2)/4 | -(√6 − √2)/4 | -(2 + √3) |
| 165° | 120° + 45° | (√6 − √2)/4 | -(√6 + √2)/4 | √3 − 2 |
How to use it
- For a textbook expansion, enter α and β (degrees or An angle unit where a full turn is 2π instead of 360 degrees.) and read all six results.
- For an exact value, switch to the second mode and type the angle; the steps show which special angles were combined.
- Open Show the work to copy the substituted formula line by line.
How to read the answer
- When α and β are both multiples of 15° or 18°, each factor in the expansion is exact and so is the result.
- Otherwise the factors are decimals and the calculator says so in a warning.
- tan(α ± β) is undefined when the The bottom number of a fraction; it says how many equal parts make one whole. 1 ∓ tan α tan β equals zero or when tan α or tan β itself is undefined.
Common mistakes and edge cases
- sin(α + β) is not sin α + sin β. Test with 30° + 60°: sin 90° = 1 but 1/2 + √3/2 ≈ 1.37.
- Mixing up the middle sign in the cosine formula.
- Using an unsimplified product: (√2/2)(√3/2) = √6/4, and the final sum (√6 + √2)/4 should be left over a single denominator.
- Angles with different units: both α and β use the unit you select.
Worked examples
45° + 30°
sin 75° exactly
sin(α + β)
(√6 + √2)/4 (0.965926)
60° − 45°
α − β = 15°
sin(α + β)
(√6 + √2)/4 (0.965926)
Radians
π/3 + π/6 = π/2
sin(α + β)
1
Decimal angles
20° + 25°
sin(α + β)
√2/2 (0.707107)
Negative angle
30° + (-90°)
sin(α + β)
-√3/2 (-0.866025)
Build 75°
sin(α + β)
(√6 + √2)/4 (0.965926)
Build 15°
sin(α + β)
(√6 − √2)/4 (0.258819)
Build 105°
sin(α + β)
(√6 + √2)/4 (0.965926)
Not a multiple of 15°
sin(α + β)
—
Frequently asked questions
What is the exact value of sin 75°?+
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4 ≈ 0.9659.
How do I find cos 15°?+
Use 45° − 30°: cos 15° = cos 45° cos 30° + sin 45° sin 30° = (√6 + √2)/4. Note the plus sign, because cos(α − β) uses a plus.
Is sin(a + b) the same as sin a + sin b?+
No. Sine is not linear. sin(30° + 60°) = 1 but sin 30° + sin 60° ≈ 1.366. Always use the identity.
What is the tangent sum formula?+
tan(α + β) = (tan α + tan β) / (1 − tan α tan β). For 45° + 30°: (1 + √3/3) / (1 − √3/3) = 2 + √3.
Why does the calculator pick 45° + 30° instead of 60° + 15°?+
It prefers building blocks whose values you already know (30°, 45°, 60°, and the axis angles) so that every factor in the expansion is exact.
Do the identities work for negative angles and radians?+
Yes. They hold for all real angles. Enter -90° or -π/2 for β and the expansion still applies.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Double Angle Formula Calculator
Compute sin 2θ, cos 2θ (all three forms), and tan 2θ from the angle itself or from a known sin θ, cos θ, or tan θ and its quadrant, with exact fractions and radicals and every step written out.
Half Angle Formula Calculator
Compute sin(θ/2), cos(θ/2), and tan(θ/2) from the angle or from a known cos θ or sin θ with its quadrant, choosing the correct sign for θ/2 and showing exact radicals and every step.
Trig Identities Calculator
Enter any angle to numerically verify the Pythagorean, quotient, reciprocal, cofunction, even-odd, and double-angle identities, with the complete identity reference table.
Trigonometric Functions Calculator
Find sin, cos, tan, csc, sec, and cot of any angle in degrees, radians, π-fractions, gradians, or turns, with exact radical values, quadrant, reference angle, coterminal angles, and a unit-circle diagram.
Unit Circle Calculator
Pick a special angle, type any angle, or go backward from a trig value to the angles that produce it. Get exact unit-circle coordinates (cos θ, sin θ), radians, all six trig functions, both solutions in one turn, and the complete special-angle table from 0° to 360°.
Eigenvalues and Eigenvectors Calculator
Find the eigenvalues and eigenvectors of a square matrix up to 8 x 8. Shows the characteristic polynomial, exact eigenvalues (including fractions, square roots, and complex pairs), eigenvector bases, multiplicities, and whether the matrix is diagonalizable.
Last updated: September 4, 2026