교과서 질문
Verify that each equation is an identity.
(sec α + csc α) (cos α - sin α) = cot α - tan α
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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 5.1.43
For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
cos² x
II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x
검증된 단계별 안내1
Recall the Pythagorean identity in trigonometry: \(\cos^{2} x + \sin^{2} x = 1\).
From this identity, express \(\cos^{2} x\) in terms of \(\sin^{2} x\) by rearranging the equation: \(\cos^{2} x = 1 - \sin^{2} x\).
Recognize that sometimes expressions in Column II might be written differently, such as \(1 - \sin^{2} x\) or other equivalent forms.
Check if the expression in Column II matches \(1 - \sin^{2} x\) or can be rewritten to this form to complete the identity with \(\cos^{2} x\).
Confirm that rewriting expressions using fundamental identities like the Pythagorean identity helps in matching equivalent expressions and completing the identity.
비슷한 문제에 대한 검증된 영상 답변:
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.
Pythagorean Identities
Pythagorean identities relate the squares of sine and cosine functions, such as sin²x + cos²x = 1. These identities are fundamental for rewriting expressions like cos²x in terms of sine or other trigonometric functions.
추천 영상:
6:25
Pythagorean Identities
Trigonometric Expression Manipulation
Manipulating trigonometric expressions involves rewriting functions using identities or algebraic techniques to simplify or match given forms. This skill is essential when matching expressions from different columns or proving identities.
추천 영상:
6:36Simplifying Trig Expressions
Understanding Function Notation and Powers
Recognizing that cos²x means (cos x)² is crucial for correctly applying identities and rewriting expressions. Misinterpreting notation can lead to errors in simplification or matching equivalent expressions.
추천 영상:
06:01i & j Notation
관련 실천
교과서 질문
The half-angle identity
tan A/2 = ± √[(1 - cosA)/(1 + cos A)]
can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity
tan A/2 = sin A/(1 + cos A)
can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)
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교과서 질문
For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
-tan x cos x
II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x
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교과서 질문
Perform each transformation. See Example 2.
Write cot x in terms of csc x.
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교과서 질문
Verify that each equation is an identity.
(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ
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교과서 질문
Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.
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