Domanda del libro di testo
Find the exact values of s in the given interval that satisfy the given condition.
[0 , 2π) ; cos² s = 1/2
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Ch. 3 - Radian Measure and The Unit Circle
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 4, Problema 77
Find each exact function value. See Example 3.
sin 5π/6
Guida verificata passo dopo passo1
Identify the angle given: \(\frac{5\pi}{6}\). This angle is in radians and is located in the second quadrant of the unit circle because it is between \(\pi/2\) and \(\pi\).
Recall that the sine function in the second quadrant is positive, so \(\sin \theta\) will be positive for \(\theta = \frac{5\pi}{6}\).
Use the reference angle to find the sine value. The reference angle for \(\frac{5\pi}{6}\) is \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
Recall the exact value of \(\sin \frac{\pi}{6}\), which is a commonly known special angle: \(\sin \frac{\pi}{6} = \frac{1}{2}\).
Since sine is positive in the second quadrant, \(\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}\).
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Unit Circle and Radian Measure
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on this circle, where the angle's measure represents the length of the arc subtended. Understanding how to locate angles like 5π/6 on the unit circle is essential for finding exact trigonometric values.
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Percorso guidato
06:11
Introduction to the Unit Circle
Sine Function on the Unit Circle
The sine of an angle in the unit circle is the y-coordinate of the corresponding point on the circle. For angles in different quadrants, sine values can be positive or negative. Knowing the sine values for common angles such as π/6, π/4, and π/3 helps in determining exact values for related angles like 5π/6.
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Percorso guidato
6:34Sine, Cosine, & Tangent on the Unit Circle
Reference Angles and Symmetry
Reference angles are the acute angles formed between the terminal side of an angle and the x-axis. They help simplify finding trigonometric values by relating them to known angles in the first quadrant. For 5π/6, the reference angle is π/6, and using symmetry properties of the unit circle allows determination of the sine value's sign and magnitude.
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Percorso guidato
5:31Reference Angles on the Unit Circle
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Find the exact values of s in the given interval that satisfy the given condition.
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Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.
s = 2.5
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