Problem 15
Question
\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ -\frac{5 \pi}{4} $$
Step-by-Step Solution
Verified Answer
The degree measure is \(-225\) degrees.
1Step 1: Recall the Conversion Formula
To convert an angle from radians to degrees, use the formula:\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]
2Step 2: Apply the Formula to the Given Radian Measure
Given the radian measure \(-\frac{5\pi}{4}\), substitute this into the formula:\[ \text{Degrees} = -\frac{5\pi}{4} \times \frac{180}{\pi} \]
3Step 3: Simplify the Expression
The \(\pi\) terms in the numerator and denominator cancel out, resulting in:\[ \text{Degrees} = -\frac{5}{4} \times 180 \]
4Step 4: Calculate the Result
Multiply \(-\frac{5}{4}\) by 180 to get:\[ \text{Degrees} = -225 \]
5Step 5: Interpret the Result
The degree measure of the angle is \(-225\) degrees, indicating it is measured in the clockwise direction from the positive x-axis.
Key Concepts
Radian to Degree ConversionNegative AnglesUnit Circle
Radian to Degree Conversion
When converting angles from radians to degrees, we use a straightforward formula that relates these two units of measurement. Radians and degrees both measure angles but in different ways. While degrees use a circle division into 360 parts, radians are based on the radius length along the circle’s circumference.
To change an angle from radians to degrees, utilize the formula:
Applying this to any radian measure will give you its value in degrees. For example, if you have an angle of \(-\frac{5\pi}{4}\) radians, plug it into the formula to convert it to degrees.
To change an angle from radians to degrees, utilize the formula:
- Degrees = Radians \( \times \frac{180}{\pi} \)
Applying this to any radian measure will give you its value in degrees. For example, if you have an angle of \(-\frac{5\pi}{4}\) radians, plug it into the formula to convert it to degrees.
Negative Angles
Negative angles are a great way to understand rotations and directions on the unit circle. A negative angle measures in the clockwise direction, as opposed to the more common counterclockwise measurement for positive angles. This concept is essential for rotating objects and in fields such as trigonometry and physics.
When you encounter a negative radian, such as \(-\frac{5\pi}{4}\), converting it to degrees also results in a negative degree measure. Therefore, \(-\frac{5}{4} \times 180 = -225\). Thus, the angle is -225 degrees. This negative sign indicates the direction of the angle measured from the positive x-axis. In practice, it means you would rotate 225 degrees in the clockwise direction.
Understanding negative angles helps when navigating circular paths or solving problems involving motion and circular geometry.
When you encounter a negative radian, such as \(-\frac{5\pi}{4}\), converting it to degrees also results in a negative degree measure. Therefore, \(-\frac{5}{4} \times 180 = -225\). Thus, the angle is -225 degrees. This negative sign indicates the direction of the angle measured from the positive x-axis. In practice, it means you would rotate 225 degrees in the clockwise direction.
Understanding negative angles helps when navigating circular paths or solving problems involving motion and circular geometry.
Unit Circle
The unit circle is a fundamental concept in trigonometry, providing a visual framework for understanding angles and their trigonometric values. It is a circle with a radius of 1 centered at the origin of a coordinate plane.
On the unit circle, each point on the circumference corresponds to an angle measure, commonly given in both radians and degrees. It covers all possible angles in the circle, making it extremely useful for visualizing angle measures like \(-\frac{5\pi}{4}\) radians or -225 degrees.
The concept of negative angles is naturally integrated on the unit circle, where moving clockwise from the positive x-axis indicates negative angle measurement. Hence, an angle of -225 degrees would point into the third quadrant of the circle.
By understanding the unit circle, students can better visualize and solve complex problems involving angles, trigonometric functions, and their applications in real-world scenarios.
On the unit circle, each point on the circumference corresponds to an angle measure, commonly given in both radians and degrees. It covers all possible angles in the circle, making it extremely useful for visualizing angle measures like \(-\frac{5\pi}{4}\) radians or -225 degrees.
The concept of negative angles is naturally integrated on the unit circle, where moving clockwise from the positive x-axis indicates negative angle measurement. Hence, an angle of -225 degrees would point into the third quadrant of the circle.
By understanding the unit circle, students can better visualize and solve complex problems involving angles, trigonometric functions, and their applications in real-world scenarios.
Other exercises in this chapter
Problem 15
Solve triangle \(A B C\). \(b=125, \quad c=162, \quad \angle B=40^{\circ}\)
View solution Problem 15
Sketch each triangle and then solve the triangle using the Law of Sines. $$ \angle B=29^{\circ}, \quad \angle C=51^{\circ}, \quad b=44 $$
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9–32 Find the exact value of the trigonometric function. $$\csc \left(-630^{\circ}\right)$$
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Solve triangle \(A B C\). \(a=65, \quad c=50, \quad \angle C=52^{\circ}\)
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