Domanda del libro di testo
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc θ - sin θ
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 5.RE.14
Work each problem.
Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.
Guida verificata passo dopo passo1
Identify the quadrant where the angle \( x \) lies. Since \( \frac{\pi}{2} < x < \pi \), \( x \) is in the second quadrant, where sine is positive and cosine is negative.
Recall the given value: \( \tan x = -\frac{5}{4} \). Use the identity \( \tan x = \frac{\sin x}{\cos x} \) to express sine and cosine in terms of a right triangle with opposite side 5 and adjacent side -4 (negative because cosine is negative in the second quadrant).
Calculate the hypotenuse \( r \) using the Pythagorean theorem: \( r = \sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \).
Find \( \sin x = \frac{\text{opposite}}{r} = \frac{5}{\sqrt{41}} \) and \( \cos x = \frac{\text{adjacent}}{r} = \frac{-4}{\sqrt{41}} \).
Use the definitions of the other trigonometric functions: \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \), \( \csc x = \frac{1}{\sin x} \), and \( \sec x = \frac{1}{\cos x} \). Substitute the values found for \( \sin x \) and \( \cos x \) to express \( \cot x \), \( \csc x \), and \( \sec x \).
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Trigonometric Ratios and Their Definitions
Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. For any angle x, tan x is the ratio of the opposite side to the adjacent side. Cotangent (cot x) is the reciprocal of tangent, cosecant (csc x) is the reciprocal of sine, and secant (sec x) is the reciprocal of cosine.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions
Using the Pythagorean Identity
The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This identity helps find missing trigonometric values when one ratio is known. By expressing sine and cosine in terms of tangent, you can use this identity to calculate csc x and sec x.
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Percorso guidato
6:25Pythagorean Identities
Determining the Sign of Trigonometric Functions Based on Quadrants
The sign of trigonometric functions depends on the quadrant where the angle lies. Since π/2 < x < π (second quadrant), sine is positive, while cosine and tangent are negative. This information is crucial to assign correct signs to cot x, csc x, and sec x after calculation.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
Find values of the sine and cosine functions for each angle measure.
2y, given sec y = -5/3, sin y > 0
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Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
For each expression in Column I, choose the expression from Column II that completes an identity.
4. cot 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
699
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Domanda del libro di testo
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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