Domanda del libro di testo
Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π
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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.8
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.
cot(-θ)/sec(-θ)
Guida verificata passo dopo passo1
Recall the definitions and properties of the trigonometric functions involved: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Also, remember the even-odd properties: \( \cos(-\theta) = \cos \theta \) (even function) and \( \sin(-\theta) = -\sin \theta \) (odd function).
Rewrite \( \cot(-\theta) \) using the definition of cotangent and the odd-even properties: \( \cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)} = \frac{\cos \theta}{-\sin \theta} = -\frac{\cos \theta}{\sin \theta} \).
Rewrite \( \sec(-\theta) \) using the definition of secant and the even property of cosine: \( \sec(-\theta) = \frac{1}{\cos(-\theta)} = \frac{1}{\cos \theta} \).
Form the quotient \( \frac{\cot(-\theta)}{\sec(-\theta)} \) by substituting the expressions from steps 2 and 3: \( \frac{-\frac{\cos \theta}{\sin \theta}}{\frac{1}{\cos \theta}} \).
Simplify the complex fraction by multiplying numerator and denominator appropriately to eliminate the quotient: \( -\frac{\cos \theta}{\sin \theta} \times \frac{\cos \theta}{1} = -\frac{\cos^2 \theta}{\sin \theta} \). This expression is now in terms of \( \sin \theta \) and \( \cos \theta \) only, with no quotients involving other trigonometric functions.
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Even and Odd Trigonometric Functions
Understanding the parity of trigonometric functions is essential for simplifying expressions with negative angles. Sine is an odd function, meaning sin(-θ) = -sin(θ), while cosine is even, so cos(-θ) = cos(θ). Secant and cotangent inherit parity from cosine and sine respectively, affecting how negative angles are handled.
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Percorso guidato
06:19
Even and Odd Identities
Trigonometric Identities for Cotangent and Secant
Cotangent and secant can be expressed in terms of sine and cosine: cot(θ) = cos(θ)/sin(θ) and sec(θ) = 1/cos(θ). Rewriting these functions helps to convert the given expression into a form involving only sine and cosine, facilitating simplification without quotients.
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Percorso guidato
3:23Secant, Cosecant, & Cotangent on the Unit Circle
Simplification of Trigonometric Expressions
After rewriting the expression in terms of sine and cosine, algebraic manipulation is used to eliminate quotients and combine terms. This involves multiplying numerator and denominator appropriately and applying fundamental identities to achieve a simplified expression solely in sin(θ) and cos(θ).
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Percorso guidato
6:36Simplifying Trig Expressions
Pratica correlata
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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Domanda del libro di testo
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°
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Domanda del libro di testo
Use the given information to find tan(x + y).
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
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Domanda del libro di testo
Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π
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Domanda del libro di testo
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°)
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