Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(\frac{7 \pi}{3}\) (b) \(-\frac{13 \pi}{60}\)

Step-by-Step Solution

Answer

(a) 420 degrees, (b) -39 degrees

Step 1: Convert radian to degree (a)

Given, \(\frac{7 \pi}{3}\) radians. To convert it to degrees, replace \(\pi\) by 180. The expression becomes \(\frac{7 \times 180}{3}\) = 420 degrees.

Step 2: Convert radian to degree (b)

Given, \(-\frac{13 \pi}{60}\) radians. To convert it to degrees, again replace \(\pi\) by 180 degree. Therefore, the expression becomes -\(\frac{13 \times 180}{60}\) = -39 degrees. As it's negative, this means the rotation is in the clockwise direction, unlike the normal anti-clockwise direction.

Key Concepts

Radian to Degree Conversion

Understanding how to convert between radians and degrees is crucial for solving many trigonometry problems. A radian is a way of measuring angles based on the radius of a circle. Specifically, one radian is the angle formed when the arc length is equal to the radius. In degree terms, there are exactly 360 degrees in a full circle, while there are \(2\pi\) radians in a full circle.
The formula to convert radians to degrees is very straightforward: multiply the radian value by \(\frac{180}{\pi}\). This is because \(\pi\) radians is equivalent to 180 degrees. Thus, the conversion factor \(\frac{180}{\pi}\) helps change the measure from radians to degrees smoothly.

• For example, to convert \(\frac{7\pi}{3}\) radians to degrees, you multiply \(\frac{7\pi}{3}\) by \(\frac{180}{\pi}\). This cancels out \(\pi\) and the calculation simplifies to \(\frac{7 \times 180}{3}\), which equals 420 degrees.

Remember that this conversion is essential for understanding situations where angle measurements are given in different units.

Degree Measure Basics

Degrees are one of the most commonly used units for measuring angles. One complete circle is divided into 360 equal parts, and each part is called a degree. This system is beneficial because many practical problems, especially those involving circles and geometric figures, use degrees to specify angles.
The degree measure is straightforward, especially when visualizing angles like we commonly encounter in regular polygons, navigation, and simple rotational movements. A right angle, for example, is 90 degrees, and this forms a fundamental building block for more complex shapes and movements.

• You often see angle values expressed in degrees when dealing with simple sketches, architectural plans, and many everyday scenarios.
• Knowing specific angle measures can help visualize the rotation direction and the extent of movement or the tilt in real-world contexts.

Understanding how degrees work and knowing how to convert other units like radians into degrees enables effective communication of measurements.

Understanding Negative Angles

Negative angles can initially seem confusing, but they play a critical role in rotational movements. An angle becomes negative when its rotation occurs in the clockwise direction, as opposed to the standard counterclockwise rotation. This concept is important when considering rotations in coordinate systems or mechanical motions.
For example, converting \(-\frac{13\pi}{60}\) radians to degrees gives us \(-39\) degrees. The negative sign indicates that the angle's rotation is clockwise. Unlike positive angles, which conventionally move counterclockwise, negative angles tell us about changes going back or in the opposite direction.

• Negative angles are commonly used in navigation or scenarios where precise directional movement is crucial, like programming rotation mechanics in animations or simulations.
• Understanding how negative angles work can help in fields ranging from geometry to physics, where direction matters.

Interpreting negative angles correctly ensures accurate representations of motions and orientations in various fields.