Trigonometric Identities Calculator

Verify and apply trigonometric identities: Pythagorean, reciprocal, quotient, co-function, and sum/difference identities. Input an angle and see all major identity values side by side.

📐 Trigonometry📐 sin²(θ) + cos²(θ) = 1🔢 Math
Angle θ (degrees)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Trigonometric Identities Calculator—sin²(θ) + cos²(θ) = 1dimensionless

Step-by-Step Examples

Example 1
Verify Pythagorean

θ=50°.

  • sin(50°)=0.766, cos(50°)=0.643
  • sin²+cos²=0.587+0.413=1.000 ✓
  • 1+tan²(50°)=sec²(50°): 2.420=2.420 ✓
✓ Identity verified
Example 2
Co-function

sin(50°)=cos(40°)?

  • sin(50°)=0.766
  • cos(90°−50°)=cos(40°)=0.766 ✓
  • Co-function: sin(θ)=cos(90°−θ)
✓ sin(50°) = cos(40°) ✓
Example 3
Double Angle

θ=30°: verify sin(60°)=2sin(30°)cos(30°).

  • 2×0.5×0.866=0.866=sin(60°) ✓
✓ Double angle identity verified ✓

Real-World Applications

📐
Trigonometry
Identities are used in every advanced trig problem and proof.
∫
Calculus
Trig identities essential for integration (e.g., sin²x=½(1−cos2x)).
🧪
Physics
Simplify expressions in oscillations, optics, quantum mechanics.
💻
Signal Processing
Fourier analysis relies on trig identities for frequency decomposition.

Common Mistakes to Avoid

⚠️
sin²+cos²=1, not sin+cos=1

The SQUARES sum to 1. sin(45°)+cos(45°)=√2 ≠ 1. Never mix this up.

⚠️
tan²+1=sec², not tan²+1=1

Three Pythagorean identities: sin²+cos²=1, tan²+1=sec², 1+cot²=csc². They are all derived from sin²+cos²=1.

⚠️
Signs in sum/difference formulas

sin(A+B)=sinAcosB+cosAsinB (plus). cos(A+B)=cosAcosB−sinAsinB (MINUS). Watch the signs carefully.

Frequently Asked Questions

What are the Pythagorean identities? ▾
sin²θ+cos²θ=1. tan²θ+1=sec²θ (divide first by cos²θ). 1+cot²θ=csc²θ (divide first by sin²θ).
What are co-function identities? ▾
sin(90°−θ)=cosθ, cos(90°−θ)=sinθ, tan(90°−θ)=cotθ. Complementary angles have swapped trig values.
What are the sum/difference identities? ▾
sin(A±B)=sinAcosB±cosAsinB. cos(A±B)=cosAcosB∓sinAsinB. Note: minus sign in cos formula when addition.
What is the product-to-sum formula? ▾
sinA×sinB=½[cos(A−B)−cos(A+B)]. Used in Fourier analysis and physics.
What are reciprocal identities? ▾
csc=1/sin, sec=1/cos, cot=1/tan. These define the three secondary trig functions.

Related Math Calculators

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Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

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