Category:
Math
All
BiologyChemistryConstructionConversionEcologyEveryday LifeFinanceFoodHealthPhysicsSportsStatisticsOther

Power Reducing Calculator

Calculate sin²(x), cos²(x), and tan²(x) using power reducing formulas

Input Parameters

Current Values

sin(x) =0.000000(0)
cos(x) =1.000000(1)
tan(x) =0.000000(0)
cos(2x) =1.000000(1)

Power Reducing Results

sin²(x)

Formula:
sin²(x) = (1 - cos(2x))/2
Result:
0.000000
Exact: 0
Verification:
Direct calculation: 0.000000
Difference: 0.00000000

cos²(x)

Formula:
cos²(x) = (1 + cos(2x))/2
Result:
1.000000
Exact: 1
Verification:
Direct calculation: 1.000000
Difference: 0.00000000

tan²(x)

Formula:
tan²(x) = (1 - cos(2x))/(1 + cos(2x))
Result:
0.000000
Exact: 0
Verification:
Direct calculation: 0.000000
Difference: 0.00000000

Step-by-Step Solution

Step 1: Calculate cos(2x)
cos(2x) = 1.000000
Step 2: Apply power reducing formulas
sin²(x) = (1 - 1.000000)/2 = 0.000000
cos²(x) = (1 + 1.000000)/2 = 1.000000
tan²(x) = (1 - 1.000000)/(1 + 1.000000) = 0.000000

Understanding Power Reducing Formulas

What are Power Reducing Formulas?

Power reducing formulas are trigonometric identities that express the squares of trigonometric functions (sin²(x), cos²(x), tan²(x)) in terms of the cosine of the double angle (cos(2x)). These formulas are essential for simplifying trigonometric expressions and solving integration problems.

The Three Main Formulas:

sin²(x) = (1 - cos(2x))/2
cos²(x) = (1 + cos(2x))/2
tan²(x) = (1 - cos(2x))/(1 + cos(2x))

Derivation from Basic Identities

These formulas are derived from the Pythagorean identity and the double angle formula:

Starting with:
sin²(x) + cos²(x) = 1
cos(2x) = cos²(x) - sin²(x)
Solving for sin²(x):
cos(2x) = 1 - 2sin²(x)
2sin²(x) = 1 - cos(2x)
sin²(x) = (1 - cos(2x))/2
Solving for cos²(x):
cos(2x) = 2cos²(x) - 1
2cos²(x) = 1 + cos(2x)
cos²(x) = (1 + cos(2x))/2

Applications and Uses

Integration

Essential for integrating powers of trigonometric functions, especially when reducing even powers to linear combinations of cosines.

Signal Processing

Used in Fourier analysis and signal processing to decompose periodic functions into simpler harmonic components.

Simplification

Helps simplify complex trigonometric expressions by reducing the power of trigonometric functions to first-order terms.