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Chapter 6: Problem 43

Convert each radian measure to degrees. $$-5 \pi$$

Short Answer

Expert verified
-900 degrees

Step by step solution

01

Understanding the Conversion Formula

The formula to convert radians to degrees is given by: \[ Degrees = Radians \times \left( \frac{180}{\pi} \right) \] This formula helps convert any radian measure into degrees.
02

Plug in the Radian Value

Substitute the given radian value \(-5 \pi\) into the conversion formula: \[ Degrees = -5 \pi \times \left( \frac{180}{\pi} \right) \]
03

Simplify the Expression

Next, simplify the equation by canceling out the common \( \pi\) terms in the numerator and denominator: \[ Degrees = -5 \times 180 \]
04

Calculate the Result

Finally, multiply the remaining numbers: \[ Degrees = -900 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

angle conversion
Angle conversion is the process of changing the measurement of an angle from one unit to another. The most common units used to measure angles are degrees and radians.
When working with angles, it's often necessary to switch between these units, especially in trigonometry and other areas of mathematics.
Understanding how to convert between degrees and radians is crucial because some mathematical formulas and functions are expressed in one unit but may need to be used in another.
For your conversion needs, remember the key formula: \( Degrees = Radians \times \frac{180}{\theta} \)
radians to degrees
The process of converting radians to degrees is straightforward.
Follow these simple steps to ensure accuracy:
  • Use the conversion formula: \( Degrees = Radians \times \frac{180}{\theta} \)
  • Substitute the given radian value into the formula.
  • Simplify the equation by canceling out common terms, usually \( \theta \) if it appears both in the numerator and the denominator.
  • Perform the arithmetic to get the final degree measure.

Let鈥檚 look at an example: Convert \(-5 \theta\) radians to degrees.
Using the formula: \( Degrees = -5 \theta \times \frac{180}{\theta} \)
Cancel out the \( \theta \): \( Degrees = -5 \times 180 \)
Final result: \( Degrees = -900 \)
So, \(-5 \theta\) radians are equal to \(-900\) degrees.
trigonometry
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is vital for various applications in science, engineering, and everyday life.
Here are some basic concepts in trigonometry:
  • Angles measured in degrees and radians.
  • The six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.
  • Conversion formulas between angles in different units.
  • Applications in finding distances, heights, and more.

Understanding trigonometry requires a solid grasp of angle measures and how to convert them. Knowing how to work with radians and degrees will help you solve many trig problems.

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Most popular questions from this chapter

Find the exact values of 6 in the given interval that satisfy the given condition. $$[-2 \pi, \pi) ; \quad 3 \tan ^{2} s=1$$

Graph each function over a one-period interval. $$y=-\frac{1}{4} \sin \left(\frac{3}{4} x+\frac{\pi}{8}\right)$$

A note on the piano has given frequency \(F\). Suppose the maximum displacement at the center of the piano wire is given by \(s(0) .\) Find constants a and \(\omega\) so that the equation $$s(t)=a \cos \omega t$$ models this displacement. Graph s in the viewing window \([0,0.05]\) by \([-0.3,0.3].\) $$F=27.5 ; s(0)=0.21$$

Find the angular speed \(\omega\) for each of the following. a line from the center to the edge of a CD revolving 300 times per min

Suppose that point \(P\) is on a circle with radius \(r,\) and ray \(O P\) is rotating with angular speed \(\omega .\) For the given values of \(r, \omega,\) and \(t,\) find each of the following. (a) the angle generated by \(P\) in time \(t\) (b) the distance traveled by \(P\) along the circle in time \(t\) (c) the linear speed of \(P\) $$r=30 \mathrm{cm}, \omega=\frac{\pi}{10} \text { radian per } \sec , t=4 \mathrm{sec}$$
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