Problem 26 - Precalculus with Calculus Previews

Question

In Problems \(25-32,\) convert the given angle from degrees to radians. $$ 15^{\circ} $$

Step-by-Step Solution

Answer

The angle \( 15^{\circ} \) is equivalent to \( \frac{\pi}{12} \) radians.

Step 1: Understand the Conversion Formula

To convert degrees to radians, we use the formula \ \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula is derived from the fact that a full circle is \( 360^{\circ} \) or \( 2\pi \) radians.

Step 2: Substitute the Given Value

Substitute the given angle in degrees into the conversion formula. In this problem, the angle is \( 15^{\circ} \). Substitute to get: \ \( 15^{\circ} \times \frac{\pi}{180} \).

Step 3: Simplify the Expression

Simplify the expression to find the angle in radians. \ First, multiply: \ \( 15 \times \frac{\pi}{180} = \frac{15\pi}{180} \). Next, reduce the fraction: \ Divide both 15 and 180 by their greatest common divisor, which is 15. \ \( \frac{15\pi}{180} = \frac{\pi}{12} \).

Step 4: Write the Final Answer

Now that the conversion is simplified, write down the final result. \ The angle \( 15^{\circ} \) is equivalent to \( \frac{\pi}{12} \) radians.

Key Concepts

Angle Measurement

Angle measurement is crucial in geometry and trigonometry. Angles can be measured in different units, the most common being degrees and radians. Understanding angle measurement is the foundation of converting between these units.
Degrees are often used in day-to-day life, such as telling time or describing directions. A full circle measures 360 degrees. Inventions like the compass rely on this measurement.
Radians, however, are the standard unit in higher-level mathematics, particularly calculus. Radians relate directly to the arc length of a circle. One full revolution around a circle is equal to \( 2\pi \) radians, not coincidentally equating numerically to \( 360^{\circ} \). This shows the direct relationship between radian and degree measurements and is why they are convertible.

Trigonometry

Trigonometry is a branch of mathematics dealing with relationships between angles and sides of triangles, particularly right triangles. Understanding radians is essential in this field, as they are the natural unit of measure for angles in calculus and other advanced studies.
Trigonometric functions, such as sine, cosine, and tangent, often use angles measured in radians for calculations. These functions help solve problems related to cycles, oscillations, and wave patterns.
Often students encounter conversions between degrees to radians within trigonometry problems. This conversion allows for smooth transitions between real-world problems, which may be described in degrees, and mathematical solutions, where radians are preferred. Thus, mastering conversion is vital in trigonometry.

Radian Measure

Radian measure is a mathematical concept that ties an angle's measure directly to the radius of a circle. One radian equals the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius.
The constant \( \pi \) plays a fundamental role here. Since the full circumference of a circle is \( 2\pi \) times the radius, a circle also has \( 2\pi \) radians. This linkage between circumference and angle measurement through radians showcases why radians are consistent across different circles, regardless of size.
To comfortably use radian measure, one should become familiar with common angle conversions:

• \( 360^{\circ} = 2\pi \) radians
• \( 180^{\circ} = \pi \) radians
• \( 90^{\circ} = \frac{\pi}{2} \) radians
• Other angles like \( 15^{\circ} \) easily convert to radians using the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This ensures precision and ease in more complex calculations.