Problem 13 - Algebra and Trigonometry

Question

Find the radian measure of the angle with the given degree measure. $$ 7.5^{\circ} $$

Step-by-Step Solution

Answer

The radian measure is \( \frac{\pi}{24} \).

Step 1: Convert Degrees to Radians Formula

To convert degrees to radians, we use the formula: \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]

Step 2: Substitute Degrees into Formula

Substitute the given degree measure (7.5°) into the formula: \[\text{Radians} = 7.5 \times \frac{\pi}{180}\]

Step 3: Simplify the Calculation

Simplify the expression by multiplying the numbers: \[\text{Radians} = \frac{7.5 \pi}{180}\]

Step 4: Reduce the Fraction

Calculate the fraction: \[\frac{7.5}{180} = \frac{75}{1800} = \frac{5}{120} = \frac{1}{24}\]. Therefore, the radian measure becomes: \[\frac{\pi}{24}\].

Key Concepts

Degrees to Radians Conversion

Many students may wonder why we need to convert degrees into radians. Degrees and radians are two units of measuring angles, just like inches and centimeters are for length. In trigonometry and other areas of mathematics, radians are often more useful because they relate directly to the geometry of a circle. A full circle is 360 degrees, but in radians, it is expressed as \(2\pi\). Therefore, the formula to convert degrees to radians is \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\). This relationship comes from the proportionality of the full circle in degrees (360°) and in radians (\(2\pi\)). Using this conversion is simple. Just take the angle in degrees and multiply it by \(\frac{\pi}{180}\). For example, converting 7.5 degrees involves multiplying it by \(\frac{\pi}{180}\), which gives the equation \(7.5 \times \frac{\pi}{180}\). This formula ensures that degrees smoothly transition to the radian format.

Fraction Simplification

Once you have set up the equation to convert degrees to radians, you might end up with a fractional expression that needs simplification. Fraction simplification is the process of reducing the fraction to its smallest possible expression. This means finding a common factor for both the numerator and the denominator. For instance, when converting 7.5 degrees to radians, the calculation \(7.5 \times \frac{\pi}{180}\) results in the fraction \(\frac{7.5 \pi}{180}\). To simplify, you match the numerator (7.5) and the denominator (180) to find their greatest common divisor. After some calculations:

• First, express 7.5 as a fraction, leading to \(\frac{75}{1800}\).
• Then, reduce this fraction step-by-step. For 75 and 1800, the greatest common divisor is 15.
• Divide both by 15 to reach \(\frac{5}{120}\), and simplifying further gives \(\frac{1}{24}\).

Thus, the simplified form of the original fraction is \(\frac{\pi}{24}\). Simplifying helps make expressions more manageable and easily interpretable.

Angle Measurement

Measuring angles is a basic aspect of geometry and trigonometry. Angles determine the direction of lines and the shape of figures. There are different units of measurements for angles—degrees and radians being the most common.Degrees are part of a system based on dividing a circle into 360 equal parts. This historical measure is easy to use in everyday contexts. Meanwhile, radians stem from the mathematical properties of the circle itself. Since a circle's circumference is \(2\pi\), one radian is the angle made by taking the radius as the arc length.
In many scientific fields, radians are preferred because they simplify mathematical expressions and calculations, particularly involving trigonometric functions. Knowing how to measure angles and convert between these units is essential for understanding a wide range of mathematical concepts, from basic geometry to advanced physics and engineering applications.