교과서 질문
In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos(β/2)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
3장, 문제 3.3.58a
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
검증된 단계별 안내1
Identify the given information: \(\sec \alpha = -3\) and the interval \(\frac{\pi}{2} < \alpha < \pi\).
Recall the relationship between secant and cosine: \(\sec \alpha = \frac{1}{\cos \alpha}\). Use this to find \(\cos \alpha\) by taking the reciprocal of \(\sec \alpha\).
Determine the sign of \(\cos \alpha\) in the given interval. Since \(\frac{\pi}{2} < \alpha < \pi\) corresponds to the second quadrant, where cosine is negative, confirm that \(\cos \alpha\) is negative.
Use the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\) to find \(\sin \alpha\). Substitute the value of \(\cos \alpha\) and solve for \(\sin^2 \alpha\).
Determine the sign of \(\sin \alpha\) in the given interval. Since sine is positive in the second quadrant, take the positive square root to find \(\sin \alpha\).
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.
Trigonometric Functions and Their Relationships
Trigonometric functions like sine, cosine, and secant describe ratios of sides in a right triangle or coordinates on the unit circle. Understanding how secant relates to cosine (sec α = 1/cos α) is essential to find sine values from given secant values.
추천 영상:
6:04
Introduction to Trigonometric Functions
Unit Circle and Angle Quadrants
The unit circle helps determine the sign and value of trig functions based on the angle's quadrant. Knowing that α lies between π/2 and π places it in the second quadrant, where sine is positive and cosine (thus secant) is negative, guiding the correct sign choice.
추천 영상:
06:11Introduction to the Unit Circle
Pythagorean Identity
The Pythagorean identity, sin²α + cos²α = 1, allows calculation of sine when cosine is known. After finding cos α from sec α, use this identity to find sin α, ensuring the correct sign based on the quadrant.
추천 영상:
6:25Pythagorean Identities
관련 실천
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, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅
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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.
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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.
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교과서 질문
In Exercises 35–38, find the exact value of the following under the given conditions:
a. sin(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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교과서 질문
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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