교과서 질문
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
cos² 5π/12
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.3.31
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
cos (x − π/2) = sin x
검증된 단계별 안내1
Start by recalling the cosine addition formula: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
In this problem, set \( a = x \) and \( b = \frac{\pi}{2} \). Substitute these values into the addition formula: \( \cos(x - \frac{\pi}{2}) = \cos x \cos \frac{\pi}{2} + \sin x \sin \frac{\pi}{2} \).
Evaluate the trigonometric functions at \( \frac{\pi}{2} \): \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \).
Substitute these values back into the equation: \( \cos(x - \frac{\pi}{2}) = \cos x \cdot 0 + \sin x \cdot 1 \).
Simplify the expression: \( \cos(x - \frac{\pi}{2}) = \sin x \), which confirms the identity.
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.
Addition Formulas
Addition formulas are trigonometric identities that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, the cosine of the difference of two angles is given by cos(a - b) = cos(a)cos(b) + sin(a)sin(b). These formulas are essential for simplifying expressions and solving trigonometric equations.
추천 영상:
05:56
Additional Rules for Indefinite Integrals
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are used to simplify expressions and solve equations in trigonometry. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities, which relate different trigonometric functions to one another.
추천 영상:
7:17Verifying Trig Equations as Identities
Co-function Identities
Co-function identities are specific trigonometric identities that relate the sine and cosine functions of complementary angles. For instance, sin(π/2 - x) = cos(x) and cos(π/2 - x) = sin(x). These identities are particularly useful when working with angles that involve π/2, as they allow for the transformation of sine and cosine functions into one another.
추천 영상:
7:17Verifying Trig Equations as Identities
관련 실천
교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = 1 - cos x
260
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교과서 질문
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the function f (x) = sin³ x.
125
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교과서 질문
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph four periods of the function f (x) = −tan 2x.
134
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교과서 질문
Can a function be both even and odd? Give reasons for your answer.
1180
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교과서 질문
A point P in the first quadrant lies on the parabola 𝔂 = 𝔁². Express the coordinates of P as functions of the angle of inclination of the line joining P to the origin.
257
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