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Chapter 6: Problem 10

Find the radian measure of the angle with the given degree measure. $$15^{\circ}$$

Short Answer

Expert verified
The radian measure for \(15^{\circ}\) is \(\frac{\pi}{12}\).

Step by step solution

01

Understanding the Conversion Factor

To convert degrees to radians, use the conversion factor \( \frac{\pi}{180^{\circ}} \). This means 180 degrees is equivalent to \( \pi \) radians.
02

Set Up the Conversion

Start by writing the formula for conversion: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}} \]. Insert \( 15^{\circ} \) for the Degrees.
03

Perform the Multiplication

Calculate \[ 15 \times \frac{\pi}{180} \]. Multiply 15 by \( \pi \) to get \( 15\pi \), and write it over 180, resulting in \( \frac{15\pi}{180} \).
04

Simplify the Fraction

Simplify \( \frac{15\pi}{180} \). Both 15 and 180 can be divided by 15, reducing the fraction to \( \frac{\pi}{12} \).

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Key Concepts

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

Angle Measurement
Angles are a fundamental concept in geometry and trigonometry. They help us understand the amount of turn between two rays that share a common endpoint, known as the vertex. In measurement, two main units are used to express angles: degrees and radians.
  • Degrees: This is perhaps the most familiar unit of angle measurement. A full circle is divided into 360 degrees, meaning one degree is a relatively small part of a circle. This scale is believed to have originated from the ancient Babylonians, who used a base-60 numeral system.
  • Radians: Unlike degrees, radians express angles based on the radius of a circle. It's an SI unit, often used in higher mathematics, like calculus. In a radian scale, a full circle is 2蟺 radians, because the circumference of a circle is 2蟺 times the radius.
Understanding angle measurement is critical as it forms the foundation for converting between units like degrees and radians.
Radian Measure
Radians offer a more natural way of expressing angles as they relate directly to circles. If you imagine the circumference of a unit circle (a circle with radius 1), the length of that circle's arc corresponding to a given angle is equal to the angle in radians.
  • One radian is the angle formed when the arc's length is equal to the circle's radius.
  • Since there are 2蟺 radians in a full circle, you can think of radians as a way to measure angles by their arc lengths.
Radians are particularly useful in mathematics involving complex functions and derivatives since they simplify many formulas and calculations. For example, the derivative of sin(x) with respect to x is cos(x) when x is measured in radians.
Conversion Factor
Converting measurements from degrees to radians involves understanding the conversion factor, which connects these two units.
  • The conversion factor is derived from the fact that 180 degrees and 蟺 radians are the same angle. Thus, the ratio used for conversion is 蟺/180.
  • To convert any angle from degrees to radians, multiply the degree measurement by this conversion factor:
    \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}}\]
During conversion, it's crucial to keep the units consistent, ensuring angles are accurately transformed from one unit to another. With a strong grasp of this conversion, you can easily switch between degrees and radians, allowing greater flexibility in solving mathematical and geometrical problems.

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Most popular questions from this chapter

\(A\) man is lying on the beach, flying a kite. He holds the end of the kite string at ground level, and estimates the angle of elevation of the kite to be \(50^{\circ} .\) If the string is \(450 \mathrm{ft}\) long, how high is the kite above the ground?

Orbit of the Earth Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. [The path of the earth around the sun is actually an ellipse with the sun at one focus (see Section 10.2 ). This ellipse, however, has very small eccentricity, so it is nearly circular. \(]\)

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\tan \theta=-4, \quad \sin \theta>0$$

\(A\) hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be \(20^{\circ}\) and \(22^{\circ} .\) How high is the balloon?

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\csc \theta=2, \quad \theta \text { in quadrant } I$$
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