Sum and Difference Identities Calculator
Calculate trigonometric sum and difference identities for sine, cosine, tangent, and more
Calculate Trigonometric Identities
Formula Used:
sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
Identity Result
Step-by-Step Calculation:
Individual Function Values:
sin(α) = 0.500000
cos(α) = 0.866025
tan(α) = 0.577350
sin(β) = 0.707107
cos(β) = 0.707107
tan(β) = 1.000000
Common Angle Examples
All Trigonometric Identities
Sum Identities (α + β)
Sine: sin(α+β) = sin(α)cos(β) + cos(α)sin(β)
Cosine: cos(α+β) = cos(α)cos(β) - sin(α)sin(β)
Tangent: tan(α+β) = (tan(α)+tan(β))/(1-tan(α)tan(β))
Results for α=30°, β=45°:
sin(30°+45°) = 0.965926
cos(30°+45°) = 0.258819
tan(30°+45°) = 3.732051
Difference Identities (α - β)
Sine: sin(α-β) = sin(α)cos(β) - cos(α)sin(β)
Cosine: cos(α-β) = cos(α)cos(β) + sin(α)sin(β)
Tangent: tan(α-β) = (tan(α)-tan(β))/(1+tan(α)tan(β))
Results for α=30°, β=45°:
sin(30°-45°) = -0.258819
cos(30°-45°) = 0.965926
tan(30°-45°) = -0.267949
Common Angle Values
Identity Tips
Sum and difference identities help calculate exact values for uncommon angles
Use special angles (30°, 45°, 60°) for exact calculations
These identities are fundamental for trigonometric proofs
Remember: cosine sum uses subtraction, difference uses addition
Understanding Sum and Difference Identities
What are Sum and Difference Identities?
Sum and difference identities are trigonometric formulas that allow us to calculate the value of a trigonometric function for the sum or difference of two angles in terms of the functions of the individual angles.
Why are they Important?
- •Calculate exact values for angles like 15°, 75°, 105°
- •Simplify complex trigonometric expressions
- •Foundation for other trigonometric identities
- •Essential for calculus and advanced mathematics
Key Memory Techniques
Sine: "Same, Same"
sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
The sign stays the same throughout
Cosine: "Opposite"
cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
The sign changes (opposite)
Tangent: "Fraction"
tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))
Signs are opposite in numerator and denominator
Example: Finding cos(15°)
Step 1: Express 15° as a difference: 15° = 45° - 30°
Step 2: Apply difference formula: cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)
Step 3: Substitute known values: = (√2/2)(√3/2) + (√2/2)(1/2)
Step 4: Simplify: = √6/4 + √2/4 = (√6 + √2)/4
Result: cos(15°) ≈ 0.9659