Problem 31 - Precalculus with Calculus Previews

Question

In Problems \(25-32,\) convert the given angle from degrees to radians. $$ -230^{\circ} $$

Step-by-Step Solution

Answer

The angle \(-230^{\circ}\) is \(-\frac{23\pi}{18}\) radians.

Step 1: Understand the Formula

To convert an angle from degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Step 2: Substitute the Degree Value

Substitute \(-230\) into the formula: \(-230 \times \frac{\pi}{180}\).

Step 3: Simplify the Expression

Simplify the multiplication: \(-230 \times \frac{\pi}{180} = -\frac{230\pi}{180}\).

Step 4: Simplify the Fraction

Reduce the fraction \(-\frac{230\pi}{180}\). The greatest common divisor of 230 and 180 is 10, so divide both by 10: \(-\frac{23\pi}{18}\).

Key Concepts

Angle Conversion

Converting angles between degrees and radians is a fundamental skill in mathematics, especially in trigonometry and calculus. Degrees and radians are both units for measuring angles, but they are used in different contexts. When you hear angles described in terms of circles and rotations, radians often provide a more natural understanding. Understanding angle conversion is essential because many mathematical functions, particularly trigonometric functions, rely on radians rather than degrees to operate correctly. For instance, when using sine, cosine, or tangent functions, radians may be the preferred unit of measure. To convert angles from degrees to radians, you'll employ a specific formula that leverages the relationship between the two measurements.

Degrees to Radians Formula

The formula to convert an angle from degrees to radians is: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]This formula is derived from the relationship that a full circle in degrees is 360°, while in radians, it is \(2\pi\). Therefore, \(\pi\) radians correspond to 180°. This relationship helps establish that each degree is equal to \(\frac{\pi}{180}\) radians. To convert degrees to radians:

• Take the angle in degrees.
• Multiply it by \(\frac{\pi}{180}\).

This formula allows for seamless conversion and is particularly useful when you're dealing with circular motion problems or in any analysis involving periodic functions.

Reducing Fractions

After applying the degrees to radians conversion formula, you often end up with a fraction that can be simplified. Reducing fractions is the process of making them simpler while keeping the same value. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).In solving math problems, reduced fractions are preferred because they make further calculations easier and are generally more visually manageable. Take, for example, the fraction \(-\frac{230\pi}{180}\). When we find the GCD of 230 and 180, which is 10, we can divide both the numerator and the denominator by this GCD:

• \(-\frac{230 \div 10 \pi}{180 \div 10} = -\frac{23\pi}{18}\).

This simplifies the fraction to a form that's easier to interpret and use in further calculations.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are crucial in various fields such as physics, engineering, and even computer graphics. These functions describe the relationships between the angles and sides of right-angled triangles but also extend to other mathematical phenomena such as waves and oscillations. Typically, trigonometric functions use angle measurements in radians. This is because radians provide a direct link between the linear measurements on circles (arc length) and the circle's radius. When you solve problems involving trigonometric functions, it’s common to convert angles from degrees to radians first. This conversion ensures more straightforward integration into the functions. For example, when calculating the sine of \( -230^\circ \), use the converted radian measure \(-\frac{23\pi}{18}\) in any calculation involving geometric or periodic analysis. Understanding and working with these functions in radians rather than degrees can often simplify calculations involving periodic trends or other phenomena dealing with changes in angle.