교과서 질문
Match each expression in Column I with its value in Column II.
(2 tan 15°)/(1 - tan² 15°)
76
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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 5.1.18
Find sinθ.
tan θ = -(√7)/2, sec θ > 0
검증된 단계별 안내1
Identify the given information: \( \tan \theta = -\frac{\sqrt{7}}{2} \) and \( \sec \theta > 0 \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so \( \sec \theta > 0 \) means \( \cos \theta > 0 \).
Determine the quadrant where \( \theta \) lies. Since \( \tan \theta \) is negative and \( \cos \theta \) is positive, \( \theta \) must be in the fourth quadrant (where cosine is positive and tangent is negative).
Use the identity relating tangent and sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Let \( \cos \theta = x \), then \( \sin \theta = \tan \theta \times x = -\frac{\sqrt{7}}{2} x \).
Apply the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = -\frac{\sqrt{7}}{2} x \) and \( \cos \theta = x \) to get \( \left(-\frac{\sqrt{7}}{2} x\right)^2 + x^2 = 1 \).
Solve the equation for \( x \) (which is \( \cos \theta \)), then use \( \sin \theta = -\frac{\sqrt{7}}{2} x \) to find \( \sin \theta \). Remember to choose the sign of \( \sin \theta \) consistent with the quadrant (fourth quadrant means \( \sin \theta < 0 \)).
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Trigonometric Ratios and Their Relationships
Trigonometric ratios like sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. Knowing one ratio, such as tangent, allows you to find others using identities or the Pythagorean theorem.
추천 영상:
6:04
Introduction to Trigonometric Functions
Sign of Trigonometric Functions in Different Quadrants
The signs of sine, cosine, and tangent depend on the quadrant where the angle lies. Given tan θ = -(√7)/2 and sec θ > 0, you can determine the quadrant by recalling that sec θ is positive where cosine is positive.
추천 영상:
6:04Introduction to Trigonometric Functions
Using Pythagorean Identities to Find Missing Ratios
Pythagorean identities like 1 + tan²θ = sec²θ help find unknown trigonometric values. By substituting the given tangent value and using the sign information, you can calculate sine θ accurately.
추천 영상:
6:25Pythagorean Identities
관련 실천
교과서 질문
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1
721
views
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.
(sec θ - 1) (sec θ + 1)
777
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교과서 질문
Verify that each equation is an identity.
(tan² α + 1)/ sec α = sec α
797
views
교과서 질문
Verify that each equation is an identity.
[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ
877
views
교과서 질문
Match each expression in Column I with its value in Column II.
cos² (π/6) - sin² (π/6)
583
views