Domanda del libro di testo
Use the given information to find the quadrant of x + y.
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
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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.40
Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π
Guida verificata passo dopo passo1
Recall the double-angle identity for cosine: \(\cos(2y) = 2\cos^2(y) - 1\).
Use the given value \(\cos(2y) = -\frac{1}{3}\) and substitute it into the identity: \(-\frac{1}{3} = 2\cos^2(y) - 1\).
Solve the equation for \(\cos^2(y)\): add 1 to both sides to get \(\frac{2}{3} = 2\cos^2(y)\), then divide both sides by 2 to find \(\cos^2(y) = \frac{1}{3}\).
Find \(\cos(y)\) by taking the square root: \(\cos(y) = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}}\). Determine the correct sign of \(\cos(y)\) using the interval \(\frac{\pi}{2} < y < \pi\), where cosine is negative.
Use the Pythagorean identity \(\sin^2(y) + \cos^2(y) = 1\) to find \(\sin(y)\): substitute \(\cos^2(y) = \frac{1}{3}\), then solve for \(\sin(y)\), considering the sign of \(\sin(y)\) in the given interval.
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Concetti chiave
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Double-Angle Identity for Cosine
The double-angle identity states that cos(2y) = 2cos²(y) - 1 or cos(2y) = 1 - 2sin²(y). This identity allows us to express cos(2y) in terms of sin(y) or cos(y), which is essential for finding sin(y) when cos(2y) is known.
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Percorso guidato
05:06
Double Angle Identities
Sign of Trigonometric Functions in Quadrants
The value of sin(y) depends on the quadrant where angle y lies. Since π/2 < y < π, y is in the second quadrant where sine is positive and cosine is negative. This information helps determine the correct sign of sin(y) after calculation.
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Percorso guidato
6:36Quadratic Formula
Pythagorean Identity
The Pythagorean identity, sin²(y) + cos²(y) = 1, relates sine and cosine of the same angle. It is useful for finding sin(y) once cos(y) is determined or vice versa, ensuring the values satisfy this fundamental trigonometric relationship.
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Percorso guidato
6:25Pythagorean Identities
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Domanda del libro di testo
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Use the given information to find tan(x + y).
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