Textbook Question
In Exercises 35–60, find the reference angle for each angle.
7𝜋/4
4
views
Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 1, Problem 42
Find the reference angle for each angle.
5π/4
Verified step by step guidance1
Understand that the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.
Since the angle is given in radians, recall that one full rotation is \(2\pi\) radians, and the quadrants are divided as follows: Quadrant I: \(0\) to \(\frac{\pi}{2}\), Quadrant II: \(\frac{\pi}{2}\) to \(\pi\), Quadrant III: \(\pi\) to \(\frac{3\pi}{2}\), Quadrant IV: \(\frac{3\pi}{2}\) to \(2\pi\).
Determine which quadrant the angle \(\frac{5\pi}{4}\) lies in by comparing it to these boundaries.
For an angle in Quadrant III, the reference angle \(\theta_{ref}\) is found by subtracting \(\pi\) from the given angle: \(\theta_{ref} = \theta - \pi\).
Apply this formula to find the reference angle for \(\frac{5\pi}{4}\) by calculating \(\frac{5\pi}{4} - \pi\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reference Angle
A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It is always positive and less than or equal to 90°, used to simplify trigonometric calculations by relating angles in different quadrants to their acute counterparts.
Recommended video:
5:31
Reference Angles on the Unit Circle
Radians and the Unit Circle
Angles can be measured in radians, where 2π radians equal 360°. The unit circle helps visualize angles and their positions in different quadrants, which is essential for determining reference angles and understanding trigonometric function values.
Recommended video:
06:11Introduction to the Unit Circle
Quadrants and Angle Positioning
The coordinate plane is divided into four quadrants, each affecting the sign and calculation of trigonometric functions. Knowing which quadrant an angle lies in helps determine how to find its reference angle by measuring the shortest distance to the x-axis.
Recommended video:
05:50Drawing Angles in Standard Position
Related Practice
Textbook Question
In Exercises 41–56, 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 41–43, find the exact value of each of the remaining trigonometric functions of θ.
cos θ = 2/5, sin θ < 0
Textbook Question
In Exercises 41–56, 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
Textbook Question
In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.
csc 17°
1
views
Textbook Question
In Exercises 41–56, 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/4