Domanda del libro di testo
Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).
<IMAGE>
sin x = ―√3/2
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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
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
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Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 3
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 3Capitolo 7, Problema 3
Which one of the following equations has solution 3π/4
a. arctan 1 = x
b. arcsin √2/2 = x
c. arccos (―√2 /2) = x
Guida verificata passo dopo passo1
Recall that the inverse trigonometric functions arctan, arcsin, and arccos return principal values within specific ranges: arctan returns values in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), arcsin returns values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and arccos returns values in \([0, \pi]\).
Check if \(x = \frac{3\pi}{4}\) lies within the principal value range of each inverse function: for arctan, \(\frac{3\pi}{4}\) is outside \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\); for arcsin, \(\frac{3\pi}{4}\) is outside \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\); for arccos, \(\frac{3\pi}{4}\) is inside \([0, \pi]\).
Evaluate the trigonometric values at \(x = \frac{3\pi}{4}\): \(\tan\left(\frac{3\pi}{4}\right)\), \(\sin\left(\frac{3\pi}{4}\right)\), and \(\cos\left(\frac{3\pi}{4}\right)\) to see which matches the given values in the equations.
Compare the given values in the problem with the values from step 3: \(\tan\left(\frac{3\pi}{4}\right)\), \(\sin\left(\frac{3\pi}{4}\right)\), and \(\cos\left(\frac{3\pi}{4}\right)\) to identify which equation has \(x = \frac{3\pi}{4}\) as a solution.
Conclude which inverse trigonometric equation corresponds to \(x = \frac{3\pi}{4}\) based on the principal value ranges and the matching trigonometric values.
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Concetti chiave
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Inverse Trigonometric Functions
Inverse trigonometric functions, such as arcsin, arccos, and arctan, return the angle whose trigonometric ratio matches a given value. They are used to find angles from known sine, cosine, or tangent values, with outputs restricted to specific principal value ranges.
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Percorso guidato
4:28
Introduction to Inverse Trig Functions
Principal Value Ranges of Inverse Functions
Each inverse trig function has a principal value range where it produces unique outputs: arcsin ranges from -π/2 to π/2, arccos from 0 to π, and arctan from -π/2 to π/2. Understanding these ranges helps determine if a given angle like 3π/4 can be a solution.
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Percorso guidato
4:22Domain and Range of Function Transformations
Evaluating Trigonometric Values at Specific Angles
Knowing the exact sine, cosine, and tangent values at common angles such as π/4, 3π/4, and π/2 is essential. For example, sin(3π/4) = √2/2, cos(3π/4) = -√2/2, and tan(3π/4) = -1, which helps identify which inverse function equation corresponds to the angle 3π/4.
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Percorso guidato
7:28Evaluate Composite Functions - Values Not on Unit Circle
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<IMAGE>
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<IMAGE>
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