Textbook Question
Find the reference angle for each angle.
5Ο/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 41
In Exercises 41β56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.
7π/6
Verified step by step guidance1
Understand that the angle is given in radians as \(\frac{7\pi}{6}\), which means the angle is measured counterclockwise from the positive x-axis.
Recall that \(\pi\) radians corresponds to 180 degrees, so \(\frac{7\pi}{6}\) is slightly more than \(\pi\) (180 degrees), specifically \(\pi + \frac{\pi}{6}\).
Since \(\pi\) radians points directly to the negative x-axis, adding \(\frac{\pi}{6}\) radians (30 degrees) moves the terminal side of the angle into the third quadrant.
Draw the angle starting from the positive x-axis, rotating counterclockwise past \(\pi\) radians, and stopping at \(\frac{7\pi}{6}\) radians, which lies in the third quadrant.
State that the angle \(\frac{7\pi}{6}\) lies in the third quadrant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angles in Standard Position
An angle is in standard position when its vertex is at the origin of the coordinate system and its initial side lies along the positive x-axis. The terminal side is determined by rotating the initial side counterclockwise for positive angles and clockwise for negative angles.
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Drawing Angles in Standard Position
Radian Measure of Angles
Radians measure angles based on the radius of a circle. One full rotation around a circle is 2Ο radians. Angles can be expressed as multiples or fractions of Ο, which allows direct use in trigonometric functions without converting to degrees.
Recommended video:
5:04Converting between Degrees & Radians
Quadrants in the Coordinate Plane
The coordinate plane is divided into four quadrants by the x- and y-axes. The quadrant in which an angle's terminal side lies depends on the angle's measure. For example, an angle between Ο and 3Ο/2 radians lies in the third quadrant.
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6:36Quadratic Formula
Related Practice
Textbook Question
In Exercises 35β60, find the reference angle for each angle.
7π/4
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Textbook Question
In Exercises 41β56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.
3π/4
Textbook Question
In Exercises 41β43, find the exact value of each of the remaining trigonometric functions of ΞΈ.
cos ΞΈ = 2/5, sin ΞΈ < 0
Textbook Question
In Exercises 33β42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
cos t + cos(t + 1000π) - tan t - tan(t + 999π) - sin t + 4 sin(t - 1000π)
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Textbook Question
In Exercises 39β48, use a calculator to find the value of the trigonometric function to four decimal places.
csc 17Β°
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