Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which formula represents cos(2θ) using sin(θ) and cos(θ)?

1. cos²θ - sin²θ

2. 2cos²θ - 1

3. 1 - 2sin²θ

4. All of the above

The correct answer is: All of the above

The correct answer is that all the provided formulas represent cos(2θ) in terms of sin(θ) and cos(θ). The double angle formula for cosine can be expressed in various forms based on the sine and cosine functions. The first formula, cos²θ - sin²θ, is derived directly from the identity, stating that the cosine of double an angle is equal to the difference between the square of the cosine and the square of the sine of that angle. The second formula, 2cos²θ - 1, represents the same idea but uses a different arrangement. This formula can be derived from the first by applying the Pythagorean identity sin²θ + cos²θ = 1, which allows sin²θ to be expressed as 1 - cos²θ. The third formula, 1 - 2sin²θ, also indicates the cosine of double an angle, this time expressed using the sine function predominantly. Similarly, it can be derived from cosine's double angle identity by manipulating the Pythagorean identity. Since all three of these formulas accurately represent cos(2θ) and can be derived from one another using fundamental trigonometric identities, the answer encompasses all the options provided,