교과서 질문
Perform each transformation. See Example 2.
Write sec x in terms of sin x.
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
6장, 문제 5.1.20
Find sinθ.
sec θ = 7/2, tan θ < 0
검증된 단계별 안내1
Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). Given \(\sec \theta = \frac{7}{2}\), find \(\cos \theta\) by taking the reciprocal: \(\cos \theta = \frac{2}{7}\).
Use the Pythagorean identity to find \(\sin \theta\): \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = \frac{2}{7}\) to get \(\sin^2 \theta = 1 - \left(\frac{2}{7}\right)^2\).
Simplify the expression for \(\sin^2 \theta\): \(\sin^2 \theta = 1 - \frac{4}{49} = \frac{49}{49} - \frac{4}{49} = \frac{45}{49}\).
Take the square root to find \(\sin \theta\): \(\sin \theta = \pm \sqrt{\frac{45}{49}} = \pm \frac{\sqrt{45}}{7}\). Simplify \(\sqrt{45}\) if desired.
Determine the correct sign of \(\sin \theta\) using the information \(\tan \theta < 0\). Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cos \theta\) is positive, \(\sin \theta\) must be negative to make \(\tan \theta\) negative.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Reciprocal Trigonometric Functions
The secant function (sec θ) is the reciprocal of the cosine function, meaning sec θ = 1/cos θ. Knowing sec θ allows you to find cos θ by taking its reciprocal, which is essential for determining sin θ using trigonometric identities.
추천 영상:
6:04
Introduction to Trigonometric Functions
Pythagorean Identity
The Pythagorean identity states that sin²θ + cos²θ = 1. Once cos θ is known, this identity helps calculate sin θ by rearranging to sin θ = ±√(1 - cos²θ). The sign depends on the quadrant where θ lies.
추천 영상:
6:25Pythagorean Identities
Sign of Trigonometric Functions by Quadrant
The sign of sine, cosine, and tangent functions depends on the quadrant of the angle θ. Given tan θ < 0 and sec θ > 0, θ lies in a quadrant where cosine is positive and tangent is negative, which helps determine the correct sign of sin θ.
추천 영상:
6:36Quadratic Formula
관련 실천
교과서 질문
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
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교과서 질문
Verify that each equation is an identity.
sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)
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교과서 질문
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
(csc θ sec θ)/cot θ
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교과서 질문
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
tan θ cos θ
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교과서 질문
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
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