Problem 12 - Precalculus: Functions and Graphs

Question

Exer. \(9-12:\) Find the exact radian measure of the angle. $$\begin{array}{llll} \text { a) } 630^{\circ} & \text { (b) } 54^{\circ} & \text { (c) } 95^{\circ} \end{array}$$

Step-by-Step Solution

Answer

a) \(\frac{7\pi}{2}\); b) \(\frac{3\pi}{10}\); c) \(\frac{95\pi}{180}\).

Step 1: Convert Degrees to Radians for Part a

To convert degrees to radians, use the formula: \[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \]For 630°, substitute in the formula:\[ 630 \times \left( \frac{\pi}{180} \right) = \frac{630\pi}{180} \]Simplify the fraction to get the radian measure: \[ \frac{630}{180} = \frac{21}{6} = \frac{7}{2} \]Thus, the exact radian measure for part a is \( \frac{7\pi}{2} \).

Step 2: Convert Degrees to Radians for Part b

Use the conversion formula for part b:\[ 54 \times \left( \frac{\pi}{180} \right) = \frac{54\pi}{180} \]Simplify the fraction:\[ \frac{54}{180} = \frac{3}{10} \]So, the exact radian measure for part b is \( \frac{3\pi}{10} \).

Step 3: Convert Degrees to Radians for Part c

For 95°, apply the conversion formula:\[ 95 \times \left( \frac{\pi}{180} \right) = \frac{95\pi}{180} \]This fraction cannot be simplified further, so the exact radian measure for part c is \( \frac{95\pi}{180} \).

Key Concepts

Degree to Radian Conversion

Converting degrees to radians is a fundamental concept in trigonometry and mathematical calculations involving angles. It involves changing the unit of measurement from degrees, which is commonly used, to radians, which is often used in calculus and higher-level math.
To perform this conversion, there's a simple formula:

• Radian Measure = Degree Measure \( \times \left( \frac{\pi}{180} \right) \)

This formula arises because a full circle is 360 degrees or \( 2\pi \) radians. Therefore, \( 180^\circ \) corresponds to \( \pi \), and the conversion factor \( \frac{\pi}{180} \) transforms a degree measure into radians. This process is used regardless of the angle size, whether it's acute, obtuse, or even a full rotation such as 630° as in our example.
When converting degrees to radians, we simply multiply the number of degrees by \( \frac{\pi}{180} \) to obtain the radian measure. This transformation is useful for simplifying complex trigonometric formulas and is important in fields such as physics, engineering, and computer science.

Angle Measurement

Understanding angle measurement is critical for solving problems in geometry and trigonometry. Angles can be measured in degrees or radians, and knowing how to work within these units is key.
Degrees are the most familiar unit of angular measurement and are commonly used in day-to-day situations. A full circle contains 360 degrees. For example, a 630-degree angle represents more than one full circle, which can be understood as a circle (360 degrees) plus 270 more degrees. This measurement method is intuitive and straightforward.
On the other hand, radians offer a measurement system that aligns more closely with the properties of a circle, making it preferred in many mathematical applications. One radian is the angle created when the radius is wrapped along the circle's edge. Thus, transforming degrees into radians involves using the relationship between the degree's proportion of a circle and its equivalent in radians.

• A 90-degree angle is \( \frac{\pi}{2} \) radians.
• A 180-degree angle is \( \pi \) radians.
• A 270-degree angle is \( \frac{3\pi}{2} \) radians.
• A 360-degree angle completes \( 2\pi \) radians.

Recognizing these equivalences helps in comparing angles and utilizing different mathematical models that require specific measurement units.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, especially when converting degrees to radians, as it makes the results clearer and more manageable. Simplification involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1.
For example, converting 630 degrees to radians gives \( \frac{630\pi}{180} \). When simplifying this fraction:

• First, find the greatest common divisor (GCD) of the numerator and the denominator.
• In this case, the GCD of 630 and 180 is 90.
• Dividing both the numerator and denominator by 90 gives \( \frac{7\pi}{2} \).

By simplifying, the fraction becomes more understandable, and handling mathematical operations with it gets easier.
Remember, not all fractions will have a simple form. For instance, \( \frac{95\pi}{180} \) cannot be simplified further because 95 and 180 have no common factors. Practicing fraction simplification builds familiarity and efficiency in mathematical operations, making studying more accessible and efficient. Remember to always check for a GCD when looking to simplify fractions.