Concept Check Suppose that ―90° \u003c θ \u003c 90° . Find the sign of each function value. cos(θ―180°)

Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 12th Edition

Ch. 1 - Trigonometric Functions

문제 103

2장, 문제 103

Concept Check Suppose that ―90° < θ < 90° . Find the sign of each function value. cos(θ―180°)

검증된 단계별 안내

Recall the given range for \( \theta \): \( -90^\circ < \theta < 90^\circ \). This means \( \theta \) is in either Quadrant I or Quadrant IV.

Rewrite the expression inside the cosine function: \( \cos(\theta - 180^\circ) \). Using the cosine subtraction formula or the cosine shift identity, recognize that \( \cos(\alpha - 180^\circ) = -\cos(\alpha) \). So, \( \cos(\theta - 180^\circ) = -\cos(\theta) \).

Determine the sign of \( \cos(\theta) \) for \( \theta \) in the interval \( (-90^\circ, 90^\circ) \). Since cosine is positive in Quadrant I (0° to 90°) and positive in Quadrant IV (-90° to 0°), \( \cos(\theta) > 0 \) in this range.

Since \( \cos(\theta) > 0 \), then \( -\cos(\theta) < 0 \). Therefore, \( \cos(\theta - 180^\circ) \) is negative for \( \theta \) in the given interval.

Summarize: The sign of \( \cos(\theta - 180^\circ) \) is negative when \( -90^\circ < \theta < 90^\circ \).

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.

Angle Measurement and Quadrants

Angles in trigonometry are measured in degrees or radians and are positioned within four quadrants on the coordinate plane. Knowing that θ is between -90° and 90° places it in Quadrants I or IV, which helps determine the sign of trigonometric functions based on the angle's location.

Reference Angles and Angle Transformations

Transforming an angle by adding or subtracting 180° shifts it to the opposite side of the unit circle. For cos(θ - 180°), the angle moves to a quadrant opposite to θ, affecting the sign of the cosine value due to the symmetry and periodicity of trigonometric functions.

Sign of the Cosine Function in Different Quadrants

The cosine function is positive in Quadrants I and IV and negative in Quadrants II and III. Understanding which quadrant the transformed angle lies in allows us to determine whether cos(θ - 180°) is positive or negative based on the cosine sign rules.