Domanda del libro di testo
Find the exact value of each real number y. Do not use a calculator.
y = sin⁻¹ √2/2
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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 4
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 4Capitolo 7, Problema 4
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
Guida verificata passo dopo passo1
Identify the trigonometric function and the given value: here, we have \( \sin x = -\frac{\sqrt{3}}{2} \).
Recall the unit circle values where \( \sin \theta = \pm \frac{\sqrt{3}}{2} \). The positive value \( \frac{\sqrt{3}}{2} \) corresponds to angles \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \), so the negative value will correspond to angles in the third and fourth quadrants.
Determine the reference angle: since \( \sin x = -\frac{\sqrt{3}}{2} \), the reference angle is \( \frac{\pi}{3} \).
Find the angles in the interval \( [0, 2\pi) \) where sine is negative with the reference angle \( \frac{\pi}{3} \). These are \( \pi + \frac{\pi}{3} \) and \( 2\pi - \frac{\pi}{3} \).
Write the solutions explicitly as \( x = \frac{4\pi}{3} \) and \( x = \frac{5\pi}{3} \), which are the values of \( x \) in the interval \( [0, 2\pi) \) satisfying the equation.
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Concetti chiave
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Unit Circle and Angle Measurement
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians (0 to 2π) or degrees (0° to 360°), and each angle corresponds to a point on the circle whose coordinates represent cosine and sine values. Understanding this helps identify angles where sine or cosine take specific values.
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Percorso guidato
06:11
Introduction to the Unit Circle
Sine Function and Its Values on the Unit Circle
The sine of an angle corresponds to the y-coordinate of the point on the unit circle at that angle. Knowing the common sine values, such as ±√3/2, and their corresponding angles allows solving equations like sin x = -√3/2 by finding all angles in the given interval with that sine value.
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Percorso guidato
6:34Sine, Cosine, & Tangent on the Unit Circle
Solving Trigonometric Equations Over a Specified Interval
When solving trigonometric equations, it is essential to find all solutions within the given domain, such as [0, 2π) for radians or [0°, 360°) for degrees. This involves identifying all angles where the trigonometric function equals the given value, considering the function’s periodicity and symmetry on the unit circle.
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Percorso guidato
4:34How to Solve Linear Trigonometric Equations
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The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹ x.
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Domanda del libro di testo
Which one of the following equations has solution 3π/4
a. arctan 1 = x
b. arcsin √2/2 = x
c. arccos (―√2 /2) = x
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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>
cos x = √3/2
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Domanda del libro di testo
Decide whether each statement is true or false. If false, explain why.
The tangent and secant functions are undefined for the same values.
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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 = ―1/2
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