Problem 17 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$315^{\circ}$$

Step-by-Step Solution

Answer

315° is \(\frac{7\pi}{4}\) radians.

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the formula: \[ ext{Radians} = ext{Degrees} imes \frac{\pi}{180} \] This formula is derived from the fact that \(180^{\circ}\) is equal to \(\pi\) radians.

Step 2: Apply the Conversion Formula

Substitute \(315^{\circ}\) into the conversion formula:\[ 315^{\circ} \times \frac{\pi}{180} \] This will convert \(315^{\circ}\) into radians.

Step 3: Perform the Multiplication

Calculate the multiplication:\[ 315 \times \frac{\pi}{180} = \frac{315\pi}{180} \] Now simplify \(\frac{315\pi}{180}\) by finding the greatest common divisor of 315 and 180.

Step 4: Simplify the Fraction

The greatest common divisor of 315 and 180 is 45. Divide both the numerator and the denominator by 45:\[ \frac{315\pi}{180} = \frac{(315 \div 45)\pi}{(180 \div 45)} = \frac{7\pi}{4} \] Thus, the angle measure in radians is \(\frac{7\pi}{4}\).

Key Concepts

Angle Measurement

When discussing angle measurement, it's crucial to understand the two primary units: degrees and radians. An angle is a measure of rotation between two rays that meet at a point, known as the vertex. Degrees are often used in various fields like geometry, navigation, and even in everyday language. A full circle is divided into 360 equal parts, each part representing one degree.

Radians, on the other hand, are used in mathematics due to their mathematical elegance. They provide a natural way of expressing angles because they relate directly to the radius of a circle. When a circle's arc length is equal to the radius, the angle is defined as 1 radian. Thus, a full circle in radians is about 6.283 radians, which is exactly \(2\pi\).

• 360 degrees = \(2\pi \) radians
• 180 degrees = \(\pi\) radians
• 90 degrees = \(\frac{\pi}{2}\) radians

Understanding these conversions helps in many fields such as trigonometry, calculus, and physics, where radians are the preferred unit due to their relation to the unit circle.

Radian Calculation

Converting degrees to radians involves using the conversion formula, which states: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]This formula arises because a full circle, which is 360 degrees, is also \(2\pi\) radians. Therefore, dividing by 180 (half of 360) gives you \(\pi\) per 180 degrees.
To convert 315 degrees to radians, you substitute 315 into the formula:

• \[ 315 \times \frac{\pi}{180} \]
• This becomes \[ \frac{315\pi}{180} \]

These steps underline the practical application of the conversion formula, showing you how to switch from a degree measurement to a radian angle easily.

Greatest Common Divisor

The greatest common divisor (GCD), also known as greatest common factor, is the largest integer that divides two or more numbers without leaving a remainder. Finding the GCD is crucial for simplifying fractions and is especially relevant when converting angles, as illustrated in simplifying the radian measure.

For the example of converting 315 degrees to radians, after multiplying, you get: \(\frac{315\pi}{180}\). To simplify, find the GCD of 315 and 180.

• Divide both numbers until you find that the GCD is 45.
• \(\frac{315}{45} = 7\)
• \(\frac{180}{45} = 4\)
• Thus, \( \frac{315\pi}{180} = \frac{7\pi}{4} \)

Understanding how to find and use the GCD not only helps in mathematical calculations but also aids in achieving the simplest form of your answer, enabling clearer communication and easier understanding of numerical relationships.