Double Angle Formulas by Tons of Concrete

Double Angle Formula for Cosine:

\[ \cos(2θ) = \cos²(θ) - \sin²(θ) \]

cos(2θ):

Alternative Form 1 (2cos²θ - 1):

Alternative Form 2 (1 - 2sin²θ):

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1. What is the Double Angle Formula for Cosine?

Definition: The double angle formula for cosine expresses cos(2θ) in terms of trigonometric functions of θ.

Purpose: These formulas are fundamental in trigonometry and are used to simplify expressions, solve equations, and in calculus.

2. How Does the Formula Work?

The primary formula is:

\[ \cos(2θ) = \cos²(θ) - \sin²(θ) \]

Alternative forms:

\[ \cos(2θ) = 2\cos²(θ) - 1 \]
\[ \cos(2θ) = 1 - 2\sin²(θ) \]

Explanation: All three forms are mathematically equivalent and can be derived from each other using the Pythagorean identity sin²θ + cos²θ = 1.

3. Importance of Double Angle Formulas

Details: These formulas are essential for:

Simplifying trigonometric expressions
Solving trigonometric equations
Calculus operations (integration, differentiation)
Physics and engineering applications

4. Using the Calculator

Tips: Enter any angle θ in degrees to see all three forms of the double angle formula for cosine calculated.

5. Frequently Asked Questions (FAQ)

Q1: Why are there multiple forms of the double angle formula?
A: Different forms are useful in different contexts. For example, when you only have cosine or sine terms in an expression.

Q2: How are these formulas derived?
A: They come from the cosine addition formula: cos(A+B) = cosAcosB - sinAsinB with A=B=θ.

Q3: Can these formulas be used for any angle?
A: Yes, they work for all real values of θ, both positive and negative.

Q4: Are there double angle formulas for sine and tangent?
A: Yes: sin(2θ) = 2sinθcosθ and tan(2θ) = 2tanθ/(1-tan²θ).

Q5: How accurate are the calculator results?
A: Results are accurate to 6 decimal places, using PHP's built-in trigonometric functions.