Multiple Choice
Evaluate each expression.
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3. Unit Circle
Reciprocal Trigonometric Functions on the Unit Circle
2:47 minutesProblem 13
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.
sec 11π/6
Verified step by step guidance1
Identify the angle given: here, the angle is \( t = \frac{11\pi}{6} \). This corresponds to a point on the unit circle.
Recall that the secant function is defined as the reciprocal of the cosine function: \( \sec t = \frac{1}{\cos t} \).
Locate the coordinates \((x, y)\) of the point on the unit circle corresponding to \( t = \frac{11\pi}{6} \). The \( x \)-coordinate represents \( \cos t \) and the \( y \)-coordinate represents \( \sin t \).
Calculate \( \sec \frac{11\pi}{6} \) by taking the reciprocal of the \( x \)-coordinate (cosine value) found in the previous step: \( \sec \frac{11\pi}{6} = \frac{1}{x} \).
Check if the cosine value is zero. If it is, then \( \sec t \) is undefined because division by zero is not possible.
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Video duration:
2mKey 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. Dividing the circle into equal arcs corresponds to specific radian values, which help locate points (x, y) representing cosine and sine of the angle.
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Introduction to the Unit Circle
Trigonometric Functions and Coordinates
On the unit circle, the x-coordinate of a point corresponds to the cosine of the angle t, and the y-coordinate corresponds to the sine of t. Other trigonometric functions like secant (sec) are defined in terms of sine and cosine; for example, sec t = 1/cos t. Understanding these relationships allows evaluation of trig functions using coordinates.
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Domain and Undefined Values of Trigonometric Functions
Some trigonometric functions are undefined at certain angles where their denominators are zero. For sec t = 1/cos t, the function is undefined when cos t = 0. Identifying these points on the unit circle is essential to determine when a function value does not exist or is undefined.
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4:22Domain and Range of Function Transformations
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