Problem 9 - Algebra and Trigonometry

Question

\(1-12\) . Find the radian measure of the angle with the given degree measure. $$ 96^{\circ} $$

Step-by-Step Solution

Answer

The angle in radians is \( \frac{8}{15} \pi \).

Step 1: Understanding Degrees to Radians Conversion

To convert an angle from degrees to radians, we use the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]where \(\pi\) is approximately 3.14159.

Step 2: Substituting Values

Substitute the given degree measure into the conversion formula. The given angle is 96 degrees, so substitute 96 for Degrees:\[ \text{Radians} = 96 \times \frac{\pi}{180} \]

Step 3: Simplifying the Expression

Simplify the fraction \( \frac{96}{180} \). Both numbers can be divided by 12:\[ \frac{96}{180} = \frac{8}{15} \]Thus, the expression becomes:\[ \text{Radians} = \frac{8}{15} \pi \]

Key Concepts

Degrees to Radians Conversion

Understanding how to convert between degrees and radians is crucial when dealing with trigonometric calculations or angle measurements. Degrees and radians are both units used to measure angles.
However, they are used in different contexts and mathematical settings. The conversion between them is straightforward when you remember the basic formula:

• Radians = Degrees × \( \frac{\pi}{180} \)

This formula is derived from the fact that a full circle is \( 360^{\circ} \), and the same circle is \( 2\pi \) radians.

• Thus, \( 360^{\circ} \) is equivalent to \( 2\pi \)
• Therefore, \( 180^{\circ} \) equals \( \pi \) radians

Any angle can be converted from degrees to radians using this relationship by multiplying the degree value by \( \frac{\pi}{180} \). When you convert an angle, make sure to simplify the fraction for a clean, precise measure of your angle in radians.

Angle Measurement

Angles are a fundamental aspect of geometry and trigonometry; they tell us about direction and rotation. They can be measured in different units, but the two most common are degrees and radians.

• Degrees: Most familiar, used in daily experiences like time and directions; there are 360 degrees in a full circle.
• Radians: A more "natural" unit in mathematics because it relates directly to the arc length of a circle.

Knowing these two units' conversion and application is essential when you are working on problems involving circular paths, oscillations, or periodic functions.
Understanding the concept of radians as a circle's arc is crucial because it allows you to understand trigonometric functions more deeply within calculus and higher mathematics:

Pi Approximation

Pi (\( \pi \)) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is infinite and non-repeating, making it a transcendental number. For practical calculations, we often use an approximate value of \( 3.14159 \).
In the context of angle conversion, \( \pi \) plays a pivotal role:

• The formula for degrees to radians conversion needs \( \pi \), emphasizing its importance in turning degrees into a mathematically robust unit like radians.
• The approximation \( \frac{\pi}{180} \) ensures that angles are accurately converted.

When working with angles mathematically, it's important to understand that \( \pi \) approximation can affect precision. Hence, using fractions of \( \pi \) (like \( \frac{8}{15}\pi \)) retains the full strength of precision which is extremely helpful in higher-level mathematics.