Problem 26 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. \(-\frac{13 \pi}{12}\)

Step-by-Step Solution

Answer

The angle is \(-195\) degrees.

Step 1: Understanding Radian Measure

Radian measure is a way to express angles using the radius of a circle. In this problem, the angle is given in radians as \(-\frac{13\pi}{12}\). Our task is to convert this radian measure into degrees.

Step 2: Convert Radians to Degrees

To convert from radians to degrees, we use the conversion factor \(\frac{180}{\pi}\). Multiply the given angle by this factor: \(-\frac{13\pi}{12} \times \frac{180}{\pi}\).

Step 3: Simplify the Expression

Start by canceling out \(\pi\) from the numerator and the denominator. This simplifies to \(-\frac{13}{12} \times 180\).

Step 4: Perform the Multiplication

Calculate \(-\frac{13}{12} \times 180\). First, simplify \(180\) over \(12\): \(180 \div 12 = 15\). Then multiply \(-13\) by \(15\), giving \(-195\).

Step 5: Conclude the Conversion

The product \(-195\) is the degree measure of the given angle. Thus, \(-\frac{13\pi}{12}\) radians is equivalent to \(-195\) degrees.

Key Concepts

Radian Measure

Radian measure is an alternative way to express the size of an angle. It's based on the radius and circumference of a circle, which makes it particularly useful in trigonometry and calculus. Unlike degrees, which are divided into 360 parts to fit a circle, radians align with the mathematical properties of a circle's radius. This unit is defined such that a full circle is equal to \(2\pi\) radians. This is because the circumference of a unit circle (circle with radius 1) is \(2\pi\). Thus,

• Half a circle, or \(\pi\) radians, equals 180 degrees.
• \(\frac{\pi}{2}\) radians is a quarter of a circle, or 90 degrees.

Feel comfortable with this connection: radians are fundamentally tied to circles, often making calculations smoother when dealing with periodic functions like sine and cosine.

Angle Conversion

Converting angles from radians to degrees is a key skill you often need in mathematics, especially when you need to compare or translate measurements. The conversion process uses an important relationship: \(180\) degrees equals \(\pi\) radians. Thus, when converting,

• Multiply the radian measure by \(\frac{180}{\pi}\) to get the measure in degrees.
• This factor is derived from the basic equivalence \(\pi \text{ radians} = 180 \text{ degrees}\).

For example, suppose you have \(-\frac{13\pi}{12}\) radians, like in our exercise. Simply multiply:\[-\frac{13\pi}{12} \times \frac{180}{\pi}\]During calculations, the \(\pi\) cancell each other, yielding a simpler fraction to work with:\[-\frac{13}{12} \times 180\]This gives you the final degree measure, showing just how seamlessly radians can be converted.

Angle Measurement

Angles can be measured in different units - primarily degrees and radians. These units help us communicate the size of an angle either through partitioning a circle into 360 parts (degrees) or using the circle's geometry (radians). Understanding how they relate is crucial.

• In degrees: This is the more traditional way of measuring angles, where a full circle is 360 degrees.
• In radians: Offers a natural fit with the circle's properties, with a full circle being \(2\pi \) radians, making mathematical calculations especially simple.

Knowing both units and how to switch between them allows flexibility. Whether in geometry, trigonometry, or physics, being able to measure accurately in the appropriate unit is an essential skill. This can be particularly useful in different contexts:

• Analyzing periodic functions, where radians might simplify computations.
• Understanding movement and rotations in physics.

Mastering these basics of angle measurement will provide a strong foundation for more advanced topics in mathematics and science.