Problem 25 - Precalculus with Calculus Previews

Question

In Problems \(25-32,\) convert the given angle from degrees to radians. $$ 10^{\circ} $$

Step-by-Step Solution

Answer

The angle is \( \frac{\pi}{18} \) radians.

Step 1: Understand the Conversion Formula

To convert degrees to radians, use the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]This formula is used because \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 2: Apply the Formula

Plug the given degree value into the conversion formula.For \(10^{\circ}\), replace \(\text{degrees}\) with 10 in the formula:\[ 10^{\circ} \times \frac{\pi}{180} \]

Step 3: Simplify the Expression

Perform the multiplication:\[ 10 \times \frac{\pi}{180} = \frac{10\pi}{180} \]Simplify the fraction by finding the greatest common divisor of 10 and 180, which is 10.

Step 4: Simplify the Fraction

Divide both the numerator and the denominator by 10:\[ \frac{10\pi}{180} = \frac{\pi}{18} \]

Key Concepts

Degrees to Radians

Angles can be measured in degrees or radians. Degrees are part of the way we typically understand angles, as in a circle which contains 360 degrees. Radians, however, are based on the radius of a circle. One full circle in radians is equal to the circumference divided by the radius, which is always approximately equal to 2\(\pi\) radians. Thus, when you convert degrees into radians, you're essentially comparing a part of the circle to its radius.
To understand this conversion, think of the unit circle, where the entire circle is considered 2\(\pi\) radians or 360 degrees. This means that \(180\) degrees equals \(\pi\) radians. Therefore, during conversion, degrees are transformed into a portion of \(\pi\) based on this equivalence.

Conversion Formula

The magic of converting degrees to radians happens with a simple, yet powerful formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]This formula is special because it directly uses the relationship where \(180\) degrees is identical to \(\pi\) radians.
For instance, if you want to convert \(10^\circ\) into radians, you replace the "degrees" in the formula with 10:

• Multiply 10 by \(\frac{\pi}{180}\).
• This gives you \(\frac{10\pi}{180}\) radians.

By multiplying the degrees by the fraction \(\frac{\pi}{180}\), you adjust the angle measure to fit the radian scale, capturing the essence of the angle in terms of the circle's radius.

Simplifying Fractions

After transforming degrees into radians, expressions often need simplification. This involves reducing the fraction to its simplest form. Simplifying fractions is like finding the shortest way to express the same quantity.
For example, after converting \(10^\circ\) to radians using our formula, we get \(\frac{10\pi}{180}\). At this stage, you should look for a common factor in the numerator and the denominator. Here, both 10 and 180 share a greatest common divisor of 10:

• Divide both the numerator and denominator by 10.
• The fraction simplifies from \(\frac{10\pi}{180}\) to \(\frac{\pi}{18}\).

Reducing fractions not only makes the expression cleaner but also helps in clearer conceptual understanding and computation. It often gives a more elegant view of the radian measure, simplifying further calculations and comparisons.