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Chapter 6: Problem 35

Convert each radian measure to degrees. $$\frac{\pi}{6}$$

Short Answer

Expert verified
30 degrees

Step by step solution

01

Understand the Conversion Factor

The conversion factor between radians and degrees is that 1 radian equals \(\frac{180}{\pi}\) degrees. This can be used to convert any angle from radians to degrees.
02

Multiply by the Conversion Factor

To convert \(\frac{\pi}{6}\) radians to degrees, multiply \(\frac{\pi}{6}\) by \(\frac{180}{\pi}\).
03

Simplify the Expression

When multiplying \(\frac{\pi}{6}\) by \(\frac{180}{\pi}\), the \(\pi\) terms cancel out, leaving: \[ \frac{\pi}{6} \times \frac{180}{\pi} = \frac{180}{6} = 30 \] degrees.

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Key Concepts

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

Understanding Angle Measures
Angle measures are essential in trigonometry and geometry since they describe the space between two intersecting lines or surfaces. We typically measure angles in degrees or radians.

A circle, for example, has 360 degrees. However, in higher mathematics, radians are often more useful. Radians are a measure based on the circle's radius. Here鈥檚 a crucial point: \[ 2\text{蟺} \text{ radians} = 360 \text{ degrees} \]

That means one complete revolution around a circle is 2蟺 radians. Hence, smaller segments can be easily calculated. One radian equals \[ \frac{180}{\text{蟺}} \text{ degrees} \] This conversion is handy when switching between radians and degrees.
Getting Familiar with Unit Circle
The unit circle is fundamental to trigonometry. It鈥檚 a circle with a radius of one unit centered at the origin of a coordinate system.

On the unit circle, each point corresponds to the angle created by drawing a line from the origin to that point. It helps us understand the relationship between angles and the coordinates of points on the circle.

For example: \[ \frac{\text{蟺}}{6} \text{ radians} = 30 \text{ degrees} \]

This means the point at \[ \frac{\text{蟺}}{6} \] radians will form an angle of 30 degrees with the positive x-axis. Knowing these relationships can help you convert angles easily and understand their position on the unit circle.
Introduction to Trigonometry
Trigonometry is the study of relationships between the angles and sides of triangles. In a right-angled triangle, we use trigonometric functions like sine, cosine, and tangent to find these relationships.

But trigonometry extends beyond triangles to circles, especially the unit circle. The coordinate \[ (x, y) \] on the unit circle helps represent the sine and cosine uniquely: \[ \text{cos}(\theta) = x \] and \[ \text{sin}(\theta) = y \]

Converting angle measures from radians to degrees or vice versa is a fundamental trigonometric skill. With these conversions, you can switch easily between different trigonometric contexts, like solving equations or analyzing waveforms.
  • Remember, always use \[ \frac{180}{\text{蟺}} \] as the factor to convert radians to degrees.

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Most popular questions from this chapter

Graph each function over a two-period interval. $$y=-1-2 \cos 5 x$$

A weight on a spring has initial position \(s(0)\) and period \(P\). (a) Find a function s given by \(s(t)=a\) cos \(\omega t\) that models the displacement of the weight. (b) Evaluate \(s(1) .\) Is the weight moving upward, downward, or neither when \(t=1 ?\) Support your results graphically or numerically. \(s(0)=-4\) in.; \(P=1.2\) sec

Graph each function over a one-period interval. $$y=3 \cos (4 x+\pi)$$

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\omega t\). Substituting \(\omega t\) for \(\theta\) converts \(s=r \theta\) to \(s=r \omega t .\) Use the formula \(s=r \omega t\) to find the value of the missing variable. $$s=\frac{3 \pi}{4} \mathrm{km}, r=2 \mathrm{km}, t=4 \mathrm{sec}$$

Work each problem. Find the radius of a circle in which a central angle of \(\frac{\pi}{6}\) radian determines a sector of area \(64 \mathrm{m}^{2}\)
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