Textbook Question
In Exercises 57โ70, find a positive angle less than or that is coterminal with the given angle. - 38๐/9
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.3.67
In Exercises 61โ86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(2๐/3)
Verified step by step guidance1
Identify the given angle: \(\frac{2\pi}{3}\). This angle is in radians and is between \(\pi/2\) and \(\pi\), which means it lies in the second quadrant.
Find the reference angle for \(\frac{2\pi}{3}\). The reference angle \(\theta_r\) in the second quadrant is calculated by subtracting the angle from \(\pi\): \(\theta_r = \pi - \frac{2\pi}{3}\).
Simplify the reference angle: \(\theta_r = \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3}\).
Recall the sine value of the reference angle \(\frac{\pi}{3}\). From the unit circle, \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
Determine the sign of sine in the second quadrant. Since sine is positive in the second quadrant, \(\sin \frac{2\pi}{3} = +\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
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5mKey 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 angles in different quadrants to their acute counterparts, allowing the use of known values for sine, cosine, and tangent.
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Reference Angles on the Unit Circle
Unit Circle and Radian Measure
The unit circle is a circle with radius 1 centered at the origin, where angles are measured in radians. Understanding the position of angles like 2ฯ/3 on the unit circle helps determine the sine and cosine values based on coordinates of points on the circle.
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06:11Introduction to the Unit Circle
Sign of Trigonometric Functions in Quadrants
The sign of sine, cosine, and tangent depends on the quadrant in which the angle lies. For example, sine is positive in the first and second quadrants and negative in the third and fourth, which is essential when using reference angles to find exact values.
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6:36Quadratic Formula
Related Practice
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In Exercises 57โ70, find a positive angle less than or that is coterminal with the given angle. 25๐ 6
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Textbook Question
In Exercises 57โ70, find a positive angle less than or that is coterminal with the given angle. 17๐ /5
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