Problem 13 - Algebra and Trigonometry

Question

In Exercises \(13-20,\) convert each angle in degrees to radians. Express your answer as a multiple of \(\pi\). $$ 45^{\circ} $$

Step-by-Step Solution

Answer

Therefore, 45 degrees is equal to \(\pi / 4\) radians.

Step 1: Write down the conversion formula

First, it's necessary to remember the conversion formula which is 1 degree = \(\pi / 180\) radians.

Step 2: Apply the conversion formula

Then, apply this formula to convert 45 degrees to radians. Multiply 45 by the conversion factor \(\pi / 180\). So, the calculation will be 45 * \(\pi / 180\).

Step 3: Calculate the value

Finally, simplify the fraction. 45 divided by 180 is 1/4, so the result should be \(\pi / 4\) radians.

Key Concepts

Degrees to Radians

When talking about angles, we often use degrees. However, in many mathematical contexts, especially in calculus, radians are preferred.
Radians provide a natural way to measure angles based on the radius of a circle.
This means that an angle measured in radians corresponds to the arc length created by that angle on a circle with a radius of one unit. Some common angles get specific radian values:

• 360° = 2π radians (a full circle)
• 180° = π radians (a straight line)
• 90° = π/2 radians (a right angle)

Conversion Formula

The key to converting degrees to radians is knowing the conversion formula.
Since 180 degrees equals π radians, we can derive the formula:\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]This formula helps you multiply the number of degrees by π/180 to find the radian measure.
It's straightforward and always works for any degree measure.
Just remember that radians often include π as part of the expression.

Simplifying Fractions

Simplifying fractions is crucial when converting angles to radians, as it ensures the answer is in its simplest form.
After applying the conversion formula, you often end up with a fraction that needs simplification.
This means dividing the numerator and the denominator by their greatest common divisor.For example, converting 45 degrees involves:

• Starting with \(45 \times \frac{\pi}{180}\)
• Simplifying to get \(\frac{\pi}{4}\)

This process makes the answer not only simpler but also more elegant in mathematical contexts.

Pi as a Mathematical Constant

Pi (π) is a fundamental constant in mathematics, representing the ratio of a circle's circumference to its diameter.
Approximately equal to 3.14159, π is a never-ending, non-repeating decimal.In many formulas, like those involving circles and trigonometry, π plays a key role:

• Perimeter of a circle: \(2\pi r\)
• Area of a circle: \(\pi r^2\)

When converting degrees to radians, π is often part of the result.
This highlights the universal nature of pi as it fits neatly into these calculations regardless of the angle size.