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Magazine Chapter 6: Problem 11
Find the reference angle for the given angle. (a) \(\frac{5 \pi}{7}\) (b) \(-1.4 \pi\) (c) 1.4
Short Answer
Expert verified
(a) \( \frac{2\pi}{7} \), (b) \( 0.4\pi \), (c) 1.4.
Step by step solution
01
Understanding Reference Angles
The reference angle is the smallest positive angle made by the terminal side of the given angle with the x-axis. Reference angles are always between 0 and \( rac{ hetap}{2} \) (90 degrees) in radians for given angles in the standard position.
02
Simplify Angle (a) \( \frac{5\pi}{7} \)
To find the reference angle, first determine the quadrant in which the angle lies. Since \( \frac{5\pi}{7} \approx 2.24 \) and is less than \( \pi \), it is in the second quadrant. The reference angle is \( \pi - \theta \). Thus, \( \theta_{reference} = \pi - \frac{5\pi}{7} = \frac{2\pi}{7} \).
03
Simplify Angle (b) \(-1.4\pi\)
Convert the negative angle to a positive angle by adding \(2\pi\). \(-1.4\pi + 2\pi = 0.6\pi\). This positive angle \(0.6\pi\) lies in the second quadrant, so the reference angle is \( \pi - 0.6\pi = 0.4\pi \).
04
Simplify Angle (c) 1.4
First, determine the equivalent angle in radians within \(0\) and \(2\pi\). Since \(1.4\) is already in the first quadrant (less than \(\pi/2 \approx 1.57\)), the reference angle is \(1.4\) itself because the angle lies in the first quadrant.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radians
Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians measure angles by the length of the arc created when an angle is formed. One full rotation around a circle equals an angle of \[2\pi \, \text{radians}\].
In the problem, angles are given in radians, which requires familiarity with this measurement.
- A half-circle rotation, or a straight angle, is \[\pi \, \text{radians}\].
- A quarter-circle, or a right angle, is \[\frac{\pi}{2} \, \text{radians}\].
- Angles are typically denoted in terms of \(\pi\), such as \(\frac{5\pi}{7}\).
In the problem, angles are given in radians, which requires familiarity with this measurement.
Quadrants
The coordinate plane is divided into four quadrants. Each quadrant signifies a section of the plane marked by positive and negative values of the x and y coordinates. Learning about them helps to determine where an angle lies when using the unit circle.
For example, in Quadrant II, sine is positive while cosine is negative. Knowing where an angle falls in the unit circle helps when calculating reference angles as shown with \(\frac{5\pi}{7}\) lying in Quadrant II.
In the exercise, identifying the quadrant helps to compute the reference angle by using the properties of radians.
- Quadrant I: Both x and y are positive.
- Quadrant II: x is negative, y is positive.
- Quadrant III: Both x and y are negative.
- Quadrant IV: x is positive, y is negative.
For example, in Quadrant II, sine is positive while cosine is negative. Knowing where an angle falls in the unit circle helps when calculating reference angles as shown with \(\frac{5\pi}{7}\) lying in Quadrant II.
In the exercise, identifying the quadrant helps to compute the reference angle by using the properties of radians.
Negative Angles
Negative angles represent a rotation in the clockwise direction in contrast to the typical counterclockwise rotation for positive angles. On the unit circle, this means moving clockwise from the positive x-axis.
- Negative angles can be converted to positive by adding \[2\pi\] repeatedly.
- For instance, \(-1.4\pi\) becomes \(0.6\pi\) when \(2\pi\) is added.
Positive Angles
Positive angles indicate a counterclockwise rotation from the positive x-axis. They are straightforward to interpret and are the standard way angles are presented.
For understanding reference angles, knowing the quadrant is essential, as shown for \(1.4\), where the angle itself serves as the reference angle due to its position in Quadrant I. Working with positive angles often requires less manipulation than negative angles, which is why recognizing the quadrant directly simplifies interpretation and solution.
- In the unit circle, starting from the positive x-axis and moving counterclockwise is how positive angles are drawn.
- A positive angle like \(1.4\) is already on the unit circle without conversion, making it easy to determine its quadrant.
For understanding reference angles, knowing the quadrant is essential, as shown for \(1.4\), where the angle itself serves as the reference angle due to its position in Quadrant I. Working with positive angles often requires less manipulation than negative angles, which is why recognizing the quadrant directly simplifies interpretation and solution.
