교과서 질문
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = ﹣3/4, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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3장, 문제 3.2.63a
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅
검증된 단계별 안내1
Identify the given information: \( \tan \alpha = \frac{3}{4} \) with \( \pi < \alpha < \frac{3\pi}{2} \), and \( \cos \beta = \frac{1}{4} \) with \( \frac{3\pi}{2} < \beta < 2\pi \).
Determine the signs of sine and cosine for angles \( \alpha \) and \( \beta \) based on their quadrant locations. Since \( \pi < \alpha < \frac{3\pi}{2} \), \( \alpha \) is in the third quadrant where sine and cosine are both negative. Since \( \frac{3\pi}{2} < \beta < 2\pi \), \( \beta \) is in the fourth quadrant where cosine is positive and sine is negative.
Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to find \( \sin \alpha \) and \( \cos \alpha \). Given \( \tan \alpha = \frac{3}{4} \), set \( \sin \alpha = 3k \) and \( \cos \alpha = 4k \) for some \( k \). Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \), then apply the correct signs based on the quadrant.
Similarly, find \( \sin \beta \) using the Pythagorean identity \( \sin^2 \beta + \cos^2 \beta = 1 \) with \( \cos \beta = \frac{1}{4} \). Determine the sign of \( \sin \beta \) based on the quadrant of \( \beta \).
Apply the cosine addition formula: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \). Substitute the values of \( \cos \alpha \), \( \cos \beta \), \( \sin \alpha \), and \( \sin \beta \) found in previous steps to express \( \cos(\alpha + \beta) \) exactly.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Sum of Angles Formula for Cosine
The sum of angles formula states that cos(α + β) = cos α cos β − sin α sin β. This identity allows you to find the cosine of the sum of two angles using the cosines and sines of the individual angles, which is essential for solving the problem.
추천 영상:
2:25
Verifying Identities with Sum and Difference Formulas
Determining Trigonometric Ratios from Given Conditions
Given tan α and the quadrant of α, you can find sin α and cos α by using the Pythagorean identity and the sign conventions of the quadrant. Similarly, knowing cos β and the quadrant of β helps determine sin β. This step is crucial to apply the sum formula correctly.
추천 영상:
6:08Evaluating Sums and Differences Given Conditions
Quadrant Sign Rules for Trigonometric Functions
The signs of sine, cosine, and tangent depend on the quadrant in which the angle lies. For example, in quadrant II, sine is positive and cosine is negative. Understanding these sign rules ensures accurate calculation of trigonometric values based on the given angle ranges.
추천 영상:
6:04Introduction to Trigonometric Functions
관련 실천
교과서 질문
In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
tan α = 4/3, 180° < α < 270°
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교과서 질문
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.
983
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교과서 질문
In Exercises 57–64, find the exact value of the following under the given conditions:
b. sin (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.
848
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교과서 질문
In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
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교과서 질문
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.
950
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