In Exercises 7–14, use the given information to find the exact value of each of the following: b. cos 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

검증된 단계별 안내

insert step 1> Identify that \( \sin \theta = \frac{15}{17} \) and \( \theta \) is in quadrant II, where sine is positive and cosine is negative.

insert step 2> Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).

insert step 3> Substitute \( \sin \theta = \frac{15}{17} \) into the identity: \( \left(\frac{15}{17}\right)^2 + \cos^2 \theta = 1 \).

insert step 4> Solve for \( \cos^2 \theta \) and then find \( \cos \theta \). Remember that \( \cos \theta \) is negative in quadrant II.

insert step 5> Use the double angle formula for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \) to find \( \cos 2\theta \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.