Domanda del libro di testo
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sec²θ - 1)/(csc²θ - 1)
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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 5.1.14
Find sinθ.
cot θ = -1/3, θ in quadrant IV
Guida verificata passo dopo passo1
Recall the definition of cotangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
Given \(\cot \theta = -\frac{1}{3}\), set \(\frac{\cos \theta}{\sin \theta} = -\frac{1}{3}\), which implies \(\cos \theta = -\frac{1}{3} \sin \theta\).
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) and substitute \(\cos \theta\) from the previous step: \(\sin^2 \theta + \left(-\frac{1}{3} \sin \theta\right)^2 = 1\).
Simplify the equation to find \(\sin^2 \theta\): \(\sin^2 \theta + \frac{1}{9} \sin^2 \theta = 1\), which combines to \(\frac{10}{9} \sin^2 \theta = 1\).
Solve for \(\sin \theta\) by isolating it and taking the square root, then determine the correct sign of \(\sin \theta\) based on the fact that \(\theta\) is in quadrant IV, where sine is negative.
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Cotangent and its Relationship to Sine and Cosine
Cotangent (cot θ) is the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, cot θ = cos θ / sin θ. Knowing cot θ allows us to express sine and cosine in terms of each other, which is essential for finding sin θ when cot θ is given.
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Percorso guidato
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Sign of Trigonometric Functions in Quadrants
The sign of sine and cosine depends on the quadrant where the angle θ lies. In quadrant IV, sine is negative and cosine is positive. This information helps determine the correct sign of sin θ after calculating its magnitude.
Video consigliato:
Percorso guidato
6:36Quadratic Formula
Pythagorean Identity
The Pythagorean identity states that sin²θ + cos²θ = 1. This relationship allows us to find one trigonometric function if the other is known, which is useful when cot θ is given and we need to find sin θ.
Video consigliato:
Percorso guidato
6:25Pythagorean Identities
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Use a half-angle identity to find each exact value.
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If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.
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Domanda del libro di testo
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
[1 - sin²(-θ)]/[1 + cot²(-θ)]
841
views
Domanda del libro di testo
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
cot² θ(1 + tan² θ)
709
views