Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)
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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.50
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°
Verified step by step guidance1
Recall that the tangent function has a period of 180°, so \( \tan(120^\circ) = \tan(120^\circ - 180^\circ) = \tan(-60^\circ) \), but it is often easier to use the reference angle in the second quadrant directly.
Identify the reference angle for 120°. Since 120° is in the second quadrant, the reference angle is \( 180^\circ - 120^\circ = 60^\circ \).
Recall the sign of tangent in the second quadrant. Tangent is negative in the second quadrant because sine is positive and cosine is negative, and \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
Use the exact value of \( \tan(60^\circ) \), which is \( \sqrt{3} \).
Combine the sign and the reference angle value to write \( \tan(120^\circ) = -\sqrt{3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 90°, finding the reference angle helps determine the trigonometric function's value by relating it to a known acute angle.
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Signs of Trigonometric Functions in Quadrants
The sign of trigonometric functions depends on the quadrant in which the angle lies. For example, tangent is positive in the first and third quadrants and negative in the second and fourth. Knowing the quadrant of 120° helps determine the sign of tan 120°.
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Exact Values of Trigonometric Functions for Special Angles
Certain angles like 30°, 45°, 60° have known exact trigonometric values expressed in simplified radical form. Using these known values for the reference angle allows calculation of the exact value of tan 120° without a calculator.
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Related Practice
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In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(-240°)
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Textbook Question
In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. cos θ = 0.4112
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In Exercises 49–59, find the exact value of each expression. Do not use a calculator. csc(-2𝜋/3)
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Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin(-𝜋/3)
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Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cot(-210°)
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