교과서 질문
In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos( β/2)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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3장, 문제 3.RE.38c
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2
검증된 단계별 안내1
Identify the given information: \( \sin \alpha = -\frac{1}{3} \) with \( \pi < \alpha < \frac{3\pi}{2} \), and \( \cos \beta = -\frac{1}{3} \) with \( \pi < \beta < \frac{3\pi}{2} \). Both angles are in the third quadrant.
Determine \( \cos \alpha \) using the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Substitute \( \sin \alpha = -\frac{1}{3} \) and solve for \( \cos \alpha \). Since \( \alpha \) is in the third quadrant, \( \cos \alpha \) will be negative.
Determine \( \sin \beta \) using the Pythagorean identity: \( \sin^2 \beta + \cos^2 \beta = 1 \). Substitute \( \cos \beta = -\frac{1}{3} \) and solve for \( \sin \beta \). Since \( \beta \) is in the third quadrant, \( \sin \beta \) will be negative.
Calculate \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} \) using the values found in previous steps.
Use the tangent addition formula to find \( \tan(\alpha + \beta) \):
\[
\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}
\]
Substitute the values of \( \tan \alpha \) and \( \tan \beta \) to express \( \tan(\alpha + \beta) \) 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 Tangent
The tangent of the sum of two angles α and β is given by tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows us to find the exact value of tan(α + β) using the individual tangents of α and β, which can be derived from their sine and cosine values.
추천 영상:
4:47
Sum and Difference of Tangent
Determining Quadrants and Sign of Trigonometric Functions
Knowing the quadrant in which an angle lies is essential to determine the signs of sine, cosine, and tangent. For example, if π < α < 3π/2, α is in the third quadrant where sine and cosine are negative, but tangent is positive. This helps correctly assign signs when calculating tan α and tan β.
추천 영상:
6:04Introduction to Trigonometric Functions
Using Pythagorean Identity to Find Missing Trigonometric Values
Given sin α or cos β, the corresponding cosine α or sine β can be found using the identity sin²θ + cos²θ = 1. This is crucial for computing tan α = sin α / cos α or tan β = sin β / cos β, enabling the use of the sum formula for tangent.
추천 영상:
6:25Pythagorean Identities
관련 실천
교과서 질문
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
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 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x = 2 cos x tan x
605
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교과서 질문
In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
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교과서 질문
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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교과서 질문
In Exercises 43–44, express each product as a sum or difference. sin 6x sin 4x
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