교과서 질문
For each expression in Column I, choose the expression from Column II that completes an identity.
2. csc 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
707
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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.30c
Use the given information to find tan(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 and the goal: We need to find \(\tan(x + y)\) given \(\sin y = -\frac{2}{3}\), \(\cos x = -\frac{1}{5}\), with \(x\) in quadrant II and \(y\) in quadrant III.
Recall the formula for the tangent of a sum: \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\).
Find \(\sin x\) using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(\cos x = -\frac{1}{5}\) and \(x\) is in quadrant II (where sine is positive), calculate \(\sin x = +\sqrt{1 - \left(-\frac{1}{5}\right)^2}\).
Find \(\cos y\) using the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). Since \(\sin y = -\frac{2}{3}\) and \(y\) is in quadrant III (where cosine is negative), calculate \(\cos y = -\sqrt{1 - \left(-\frac{2}{3}\right)^2}\).
Calculate \(\tan x = \frac{\sin x}{\cos x}\) and \(\tan y = \frac{\sin y}{\cos y}\), then substitute these values into the formula \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\) to find the expression for \(\tan(x + y)\).
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.
Trigonometric Identities for Sum of Angles
The tangent of a sum, tan(x + y), can be found using the identity tan(x + y) = (tan x + tan y) / (1 - tan x * tan y). This formula allows combining the tangents of individual angles to find the tangent of their sum.
추천 영상:
2:25
Verifying Identities with Sum and Difference Formulas
Determining Signs of Trigonometric Functions in Quadrants
The signs of sine, cosine, and tangent depend on the quadrant of the angle. In quadrant II, sine is positive and cosine is negative; in quadrant III, both sine and cosine are negative. Correct sign assignment is crucial for accurate calculations.
추천 영상:
6:04Introduction to Trigonometric Functions
Finding Missing Trigonometric Ratios Using Pythagorean Identity
Given one trigonometric ratio (like sin y or cos x), the other ratios can be found using the Pythagorean identity sin²θ + cos²θ = 1. This helps determine unknown values such as cos y or sin x, essential for calculating tan x and tan y.
추천 영상:
6:25Pythagorean Identities
관련 실천
교과서 질문
Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.
sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV
846
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교과서 질문
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
-sin 35°
617
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교과서 질문
Use the given information to find the quadrant of x + y.
sin y = - 2/3, cos x = -1/5 , x in quadrant II, y in quadrant III
747
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교과서 질문
Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π
828
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교과서 질문
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
cos(-55°)
741
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