Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One key identity is the double angle formula for cosine, which states that cos(2θ) can be expressed as cos²(θ) - sin²(θ) or 2cos²(θ) - 1 or 1 - 2sin²(θ). Understanding these identities is crucial for simplifying and calculating trigonometric expressions.
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Quadrants and Angle Properties
The unit circle is divided into four quadrants, each with distinct properties regarding the signs of sine and cosine. In quadrant II, sine is positive while cosine is negative. This knowledge is essential for determining the values of trigonometric functions based on the angle's location, which directly impacts the calculation of cos(2θ) in this problem.
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Finding Exact Values of Trigonometric Functions
To find the exact value of trigonometric functions, one often uses known values or relationships between the functions. Given sin(θ) = 15/17, we can find cos(θ) using the Pythagorean identity sin²(θ) + cos²(θ) = 1. This allows us to compute cos(2θ) accurately by substituting the values derived from sin(θ) and cos(θ) into the double angle formula.
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