Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$270^{\circ}$$

Step-by-Step Solution

Answer

The conversion of \(270^{\circ}\) to radians is \(\frac{3\pi}{2}\).

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 \( \pi \) radians is equivalent to \(180^{\circ}\).

Step 2: Apply the Conversion Formula

Substitute \(270^{\circ}\) into the conversion formula:\[270^{\circ} \times \frac{\pi}{180}\]

Step 3: Simplify the Expression

To simplify \(270 \times \frac{\pi}{180}\), first calculate the fraction:\[\frac{270}{180}\]Divide both the numerator and the denominator by their greatest common divisor, which is 90:\[\frac{270 \div 90}{180 \div 90} = \frac{3}{2}\]

Step 4: Finalize the Radian Measure

Substitute back into the expression for radians:\[\frac{3}{2} \times \pi\]Thus, the equivalent radian measure is \(\frac{3\pi}{2}\).

Key Concepts

Degree to Radian Conversion

The concept of degree to radian conversion is fundamental in trigonometry and calculus. Understanding how to convert between these two units of measuring angles is key to solving many mathematical problems.
To transform degrees into radians, we use a very handy formula:

• The formula is: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
• Here, \( \pi \) is a constant approximately equal to 3.14159, and 180 degrees is the number of degrees contained in a straight angle.

Substituting into this formula will give you the radian equivalent of your degree measure. Remember, multiplying by \( \frac{\pi}{180} \) changes degrees into an arc's radian length on a unit circle.
Take care to simplify your expression by reducing fractions wherever possible, just like simplifying \( \frac{270}{180} \) to \( \frac{3}{2} \) in our exercise.

Radians and Degrees

Radians and degrees are both measurements of angles, but they come from different mathematical traditions. Understanding the relationship between them is crucial for working across different mathematical contexts.
In the degree system, a full circle is 360 degrees. However, in the radian system, a full circle is measured as \(2\pi\) radians.

• One radian is the angle created from the center of a circle by an arc whose length is equal to the circle's radius.
• This makes \(\pi\) radians equivalent to 180 degrees, establishing the fundamental connection:
• \(1\text{ radian} = \frac{180}{\pi}\) degrees

Radians offer a direct measure related to the radius and arc length, providing a 'natural' unit in many mathematical and physical formulas.

Mathematics Conversion Formulae

Conversion formulae in mathematics allow us to switch between different units with ease and accuracy. The degree to radian conversion is one of these essential formulae.
Mathematically, any conversion factor like \( \frac{\pi}{180} \) can be thought of as a scale to change from one unit to another.

• This specific conversion stems from the geometric properties of circles.
• To apply it, multiply the degree measure by this factor: \( \text{degrees} \times \frac{\pi}{180} \).
• Implement this by inserting your specific degree measure, such as 270 in our example, to find the radian equivalent.

Applying these mathematical conversion formulae is straightforward, but precision is crucial. Always simplify the resulting radians where possible to make calculations cleaner and your results more readable.