Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sec 7𝜋 / 4
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.RE.49
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin 240°
Verified step by step guidance1
Recognize that the angle 240° is in the third quadrant of the unit circle, where sine values are negative.
Find the reference angle for 240° by subtracting 180°: \(240^\circ - 180^\circ = 60^\circ\).
Recall the sine value of the reference angle 60°, which is \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).
Since 240° is in the third quadrant where sine is negative, apply the sign: \(\sin 240^\circ = -\sin 60^\circ\).
Write the exact value as \(\sin 240^\circ = -\frac{\sqrt{3}}{2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Angle Measurement
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles in trigonometry are often measured in degrees or radians, and their position on the unit circle determines the values of sine and cosine. Understanding how to locate 240° on the unit circle is essential for finding sin 240°.
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Introduction to the Unit Circle
Reference Angles
A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. For angles greater than 180°, like 240°, the reference angle helps simplify the calculation of sine and cosine by relating them to known values of acute angles.
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5:31Reference Angles on the Unit Circle
Sign of Trigonometric Functions in Quadrants
The sign of sine and cosine depends on the quadrant in which the angle lies. Since 240° is in the third quadrant, where sine values are negative, this knowledge helps determine the correct sign of sin 240° after using the reference angle.
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6:36Quadratic Formula
Related Practice
Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos (11𝜋 / 6)
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Textbook Question
Use the unit circle shown to find the value of the trigonometric function.
sin (2𝜋/3)
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Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos(-35𝜋 / 6)
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Textbook Question
Use the unit circle shown to find the value of the trigonometric function.
tan 11𝜋/6
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Textbook Question
The unit circle has been divided into eight equal arcs, corresponding to t-values of
0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
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sin 3𝜋/4