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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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.3.73
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(-240°)
Verified step by step guidance1
Identify the given angle: \(-240^\circ\). Since it is negative, find its positive coterminal angle by adding \(360^\circ\): \(-240^\circ + 360^\circ = 120^\circ\).
Determine the reference angle for \(120^\circ\). Since \(120^\circ\) is in the second quadrant, the reference angle is \(180^\circ - 120^\circ = 60^\circ\).
Recall the sign of sine in the second quadrant. Sine is positive in the second quadrant, so \(\sin(120^\circ) = +\sin(60^\circ)\).
Use the exact value of \(\sin(60^\circ)\), which is \(\frac{\sqrt{3}}{2}\).
Therefore, \(\sin(-240^\circ) = \sin(120^\circ) = +\frac{\sqrt{3}}{2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reference Angles
A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to a corresponding angle in the first quadrant, where values are easier to determine.
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Reference Angles on the Unit Circle
Trigonometric Function Signs in Quadrants
The sign of sine, cosine, and tangent functions depends on the quadrant in which the angle's terminal side lies. For sine, it is positive in the first and second quadrants and negative in the third and fourth quadrants, which is essential for determining the exact value.
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Evaluating Sine of Negative Angles
The sine function is odd, meaning sin(-θ) = -sin(θ). This property allows converting negative angles into positive ones by changing the sign of the sine value, simplifying the evaluation without a calculator.
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Related Practice
Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°
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Textbook Question
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. tan θ = 4.6252
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Textbook Question
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
If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).
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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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