Sum & Difference Identities Calculator — sin(A±B), cos(A±B), tan(A±B)

About the Sum & Difference Identities Calculator — sin(A±B), cos(A±B), tan(A±B)

The sum and difference identities are among the most important formulas in trigonometry. They express sin(A ± B), cos(A ± B), and tan(A ± B) in terms of the individual trig functions of A and B. These identities are the foundation for dozens of other results in trigonometry, including double-angle formulas, half-angle formulas, product-to-sum identities, and the harmonic addition theorem.

By decomposing compound angles into known components — typically 30°, 45°, and 60° — you can find exact values for angles like 15°, 75°, 105°, and others not on the standard unit circle. This technique is essential for precalculus exams, SAT/ACT preparation, and any context where exact symbolic answers are preferred over decimal approximations.

This calculator takes two angles A and B (in degrees or radians), computes all six identities simultaneously, shows the intermediate trig values with exact forms where available, provides a step-by-step derivation for sin(A + B), and visualizes all six results on a signed bar chart.

When This Page Helps

Computing all six sum and difference identities by hand involves 12 individual trig function evaluations and careful arithmetic with irrational numbers. This calculator does the full set at once, shows exact forms for special angles, provides step-by-step verification, and visualizes the results.

It is indispensable for trigonometry students who need to practice these identities, check their homework, or explore patterns across different angle pairs.

How to Use the Inputs

1. Select the angle unit: degrees or radians.
2. Enter angle A and angle B.
3. Or click a preset for common angle decompositions like 75° = 45° + 30°.
4. View all six sum and difference identity results in the output cards.
5. Check the intermediate trig values table for sin/cos/tan of A and B with exact forms.
6. Follow the step-by-step derivation for sin(A + B).
7. Review the visual bar chart comparing all six results.

Example Calculation

Tips & Best Practices

• Common decompositions: 15° = 45°−30°, 75° = 45°+30°, 105° = 60°+45°, 165° = 180°−15°.
• The cosine sum formula has a MINUS sign (cos A cos B − sin A sin B) — the opposite of the sign in the angle.
• For the tangent identity, watch the denominator: if it's zero, the result is undefined (vertical asymptote).
• Exact values from sum identities often involve nested radicals like (√6 ± √2)/4.
• The double-angle formulas are just the sum formulas with B = A.
• To verify your result, check that sin²(A+B) + cos²(A+B) = 1 using the computed values.

Proof Using the Unit Circle

The most geometric proof of sin(A+B) uses the unit circle. Draw angle A from the positive x-axis, then rotate an additional angle B. The resulting point has coordinates (cos(A+B), sin(A+B)). By expressing this rotation as a composition of two rotations — which is equivalent to multiplying rotation matrices — you obtain the familiar formulas.

The rotation matrix approach generalizes to higher dimensions and is the foundation of computer graphics transformations, robotics kinematics, and quantum mechanics state rotations.

Historical Context

The sum and difference formulas were known (in geometric form) to the ancient Greek astronomer Ptolemy, who used them to compute his table of chords — the precursor to modern trigonometric tables. The Islamic mathematician al-Battani refined these into the forms we use today. In the Renaissance, these identities were called "prosthaphaeresis" formulas and were used as a computational technique for multiplication before logarithms were invented: to multiply two numbers, convert them to cosines, use the product-to-sum identity, and convert back.

Applications Beyond Mathematics

In signal processing, the sum identity explains how mixing two frequencies produces sum and difference frequencies (heterodyning). In physics, the addition of velocities in special relativity uses a formula analogous to the tangent sum identity: v = (v₁ + v₂)/(1 + v₁v₂/c²). In music, beat frequencies arise from cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2) — a direct consequence of these identities.

Frequently Asked Questions

• Why are sum and difference identities important?

  They are the building blocks for virtually all other trig identities (double-angle, half-angle, product-to-sum, etc.) and are crucial for solving trig equations, simplifying expressions, and computing exact values for non-standard angles.

• How do I find exact value of sin(75°)?

  Write 75° = 45° + 30°, then: sin(75°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

• When is tan(A + B) undefined?

  tan(A + B) is undefined when the denominator 1 − tan A tan B = 0, which occurs when tan A tan B = 1. For example, tan(45° + 45°) = tan 90° is undefined.

• How are these related to Euler's formula?

  Euler's formula gives e^(i(A+B)) = e^(iA) · e^(iB). Expanding both sides and equating real and imaginary parts directly produces the sum identities for cos and sin.

• Can I use these in reverse to factor sums of trig functions?

  Yes! The product-to-sum and sum-to-product identities are derived by adding or subtracting the sum/difference formulas. For example, sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2).

• Do these work for complex angles?

  Yes. The sum and difference identities hold for all complex numbers A and B, not just real angles. This is a consequence of their derivation from Euler's formula.