Multiple Choice
Evaluate each expression.
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3. Unit Circle
Reciprocal Trigonometric Functions on the Unit Circle
3:15 minutesProblem 11
Textbook Question
In Exercises 5β18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of
0, π, π, π, 2π, 5π, π, 7π, 4π, 3π, 5π, 11π, and 2π.
6 3 2 3 6 6 3 2 3 6
Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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In Exercises 11β18, continue to refer to the figure at the bottom of the previous page.
csc 7π/6
Verified step by step guidance1
Identify the angle given: \(t = \frac{7\pi}{6}\). This angle is in radians and corresponds to a point on the unit circle.
Recall that the cosecant function is the reciprocal of the sine function, so \(\csc t = \frac{1}{\sin t}\).
Locate the coordinates of the point on the unit circle corresponding to \(t = \frac{7\pi}{6}\). The coordinates are \((x, y) = (\cos t, \sin t)\).
Find the \(y\)-coordinate (which is \(\sin \frac{7\pi}{6}\)) from the unit circle. This value will be used to calculate \(\csc \frac{7\pi}{6}\).
Calculate \(\csc \frac{7\pi}{6}\) by taking the reciprocal of \(\sin \frac{7\pi}{6}\), i.e., \(\csc \frac{7\pi}{6} = \frac{1}{\sin \frac{7\pi}{6}}\). If \(\sin \frac{7\pi}{6} = 0\), then \(\csc \frac{7\pi}{6}\) is undefined.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Radian Measure
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, where one full rotation equals 2Ο radians. Understanding how to locate an angle t on the unit circle is essential for determining the corresponding coordinates (x, y) that represent cosine and sine values.
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Introduction to the Unit Circle
Trigonometric Functions and Their Coordinates
On the unit circle, the x-coordinate corresponds to cos(t) and the y-coordinate corresponds to sin(t). Other trigonometric functions like cosecant (csc t) are defined in terms of sine, with csc t = 1/sin t. Knowing how to use the coordinates to find these values is crucial, especially to identify when functions are undefined (e.g., when sin t = 0).
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Evaluating Trigonometric Functions at Specific Angles
To evaluate functions like csc(7Ο/6), first locate the angle 7Ο/6 on the unit circle, find the sine value from the y-coordinate, and then compute the reciprocal for cosecant. Recognizing common reference angles and their sine values helps simplify this process and determine if the function is defined or undefined.
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