Problem 22 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(-15^{\circ}\)

Step-by-Step Solution

Answer

\(-15^{\circ}\) equals \(-\frac{\pi}{12}\) radians.

Step 1: Understand the Conversion Relationship

Degrees and radians are two different units for measuring angles. The relationship between them is given by the formula: \[ 180^{\circ} = \pi \text{ radians} \].We can use this relationship to convert between degrees and radians.

Step 2: Set Up the Conversion

To convert from degrees to radians, we use the formula:\[ \text{radians} = \frac{\text{degrees} \times \pi}{180} \].In this case, we want to convert \(-15^{\circ}\) to radians.

Step 3: Perform the Conversion

Substitute \(-15\) into the conversion formula:\[ \text{radians} = \frac{-15 \times \pi}{180} \].Simplify the fraction:\[ \text{radians} = -\frac{15\pi}{180} \].Reduce the fraction by dividing both the numerator and the denominator by 15:\[ \text{radians} = -\frac{\pi}{12} \].

Key Concepts

Degrees to Radians

When we talk about converting degrees into radians, we are essentially switching the units of angle measurement from one system to another. This is similar to converting feet to meters in length measurements. The degree and radian are both units that can describe how large an angle is, but they do so in different ways.

To convert degrees to radians, we use the formula:

• radians = \( \frac{\text{degrees} \times \pi}{180} \)

The reason for this formula lies in the essential relationship that \( 180^{\circ} \) equals \( \pi \) radians. By using this formula, we can translate any angle given in degrees into radians, simplifying many mathematical and real-world applications where radian measure is more practical.

Specifically, in converting \(-15^{\circ}\) to radians, the calculation performed is:\(\frac{-15 \times \pi}{180} = -\frac{\pi}{12} \).

Angle Measurement

Angle measurement is a crucial concept across various fields such as physics, engineering, and even computer graphics. It refers to the size of an angle, which can be thought of as the amount of rotation needed for one arm of the angle to lie on top of the other.

There are several units used to measure angles, with degrees and radians being the most common.

• Degrees: A circle is divided into 360 degrees, making one degree a very fine unit of measurement, ideal for navigation and simple geometry problems.
• Radians: Instead, a circle is considered as \(2\pi\) radians, where one radian is the angle formed when the radius is wrapped around the circle's circumference. This is a more natural unit mathematically.

Using radians is especially common in higher mathematics, as they provide simpler formulas for derivatives and integrals of trigonometric functions.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental tools in trigonometry that relate the angles of a triangle to the lengths of its sides. They are extensively used in many areas of science and engineering, from calculating distances to modeling waves.

Understanding how these functions operate with angles measured in both degrees and radians is essential. Radians are often preferred, especially in calculus, due to their natural association with the unit circle. The behaviors of these trigonometric functions are deeply connected to angle measurements, making radians more efficient for computational purposes.

• Sine and Cosine functions cyclically repeat every \(2\pi\) radians.
• Tangent functions repeat every \(\pi\) radians.

Using radians can simplify solving complex mathematical problems, particularly those involving periodic phenomena like sound waves or alternating current.