교과서 질문
Verify that each equation is an identity.
(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
838
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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TOC.tableOfContents
- 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.RE.30a
Use the given information to find sin(x + y).
sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III
검증된 단계별 안내1
Identify the given information: \(\sin y = -\frac{2}{3}\), \(\cos x = -\frac{1}{5}\), with \(x\) in quadrant II and \(y\) in quadrant III.
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos y\). Since \(\sin y = -\frac{2}{3}\), calculate \(\cos y = \pm \sqrt{1 - \sin^2 y} = \pm \sqrt{1 - \left(-\frac{2}{3}\right)^2}\), and determine the correct sign based on the quadrant of \(y\).
Similarly, find \(\sin x\) using \(\cos x = -\frac{1}{5}\). Calculate \(\sin x = \pm \sqrt{1 - \cos^2 x} = \pm \sqrt{1 - \left(-\frac{1}{5}\right)^2}\), and choose the sign according to the quadrant of \(x\).
Recall the angle addition formula for sine: \(\sin(x + y) = \sin x \cos y + \cos x \sin y\).
Substitute the values of \(\sin x\), \(\cos y\), \(\cos x\), and \(\sin y\) into the formula to express \(\sin(x + y)\) in terms of known quantities.
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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.
Sum of Angles Formula for Sine
The sine of the sum of two angles, sin(x + y), can be found using the formula sin(x + y) = sin x cos y + cos x sin y. This identity allows us to express sin(x + y) in terms of the sines and cosines of the individual angles x and y.
추천 영상:
2:25
Verifying Identities with Sum and Difference Formulas
Determining Signs of Trigonometric Functions by Quadrant
The signs of sine and cosine depend on the quadrant in which the angle lies. In quadrant II, sine is positive and cosine is negative; in quadrant III, both sine and cosine are negative. This helps determine the correct values and signs of unknown trigonometric functions.
추천 영상:
6:04Introduction to Trigonometric Functions
Using Pythagorean Identity to Find Missing Values
The Pythagorean identity, sin²θ + cos²θ = 1, allows us to find an unknown sine or cosine value when the other is given. By substituting the known value and solving, we can find the missing trigonometric function needed to apply the sum formula.
추천 영상:
6:25Pythagorean Identities
관련 실천
교과서 질문
Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
csc x - cot x
721
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교과서 질문
For each expression in Column I, choose the expression from Column II that completes an identity.
6. sec² 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
726
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교과서 질문
Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(cos x sin 2x)/1 + cos 2x)
765
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교과서 질문
Find values of the sine and cosine functions for each angle measure.
B, given cos 2B = 1/8 , 540° < 2B < 720°
725
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교과서 질문
Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.
763
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