Problem 26 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$-320^{\circ}$$

Step-by-Step Solution

Answer

The answer is \(-\frac{16\pi}{9}\) radians.

Step 1: Understand the conversion formula

To convert degrees into radians, use the formula: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\] This formula arises because 180 degrees is equal to \(\pi\) radians. Thus, each degree equals \(\frac{\pi}{180}\) radians.

Step 2: Substitute the given degree measure

Substitute \(-320^{\circ}\) into the conversion formula: \[\text{radians} = -320 \times \frac{\pi}{180}\]

Step 3: Simplify the expression

To simplify the fraction \(-320 \times \frac{\pi}{180}\), first simplify \(\frac{-320}{180}\). Find the greatest common divisor of 320 and 180, which is 20. So, divide both the numerator and the denominator by 20: \[\frac{-320}{180} = \frac{-16}{9}\]Then multiply by \(\pi\): \[\text{radians} = \frac{-16}{9} \pi\]

Step 4: Write the final answer

The angle \(-320^{\circ}\) converted to radians is \[-\frac{16\pi}{9}\] radians.

Key Concepts

Degrees to Radians

Converting angles from degrees to radians is a common task in mathematics and physics, particularly when dealing with circular motion or trigonometry. The relationship between degrees and radians is grounded in the geometry of a circle.

• A full circle is composed of 360 degrees.
• The same circle is also defined to have an angle of \(2\pi\) radians.
• This means that 180 degrees is equivalent to \(\pi\) radians.

To convert any angle in degrees into radians, you multiply the degrees by \(\frac{\pi}{180}\). This simple multiplication converts the angular measure from one unit to another while maintaining the same proportion. Understanding this relationship allows for smooth transitions between the two measurement systems without error.

Conversion Formula

The cornerstone of converting degrees to radians is the conversion formula: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]Here's how it works:

• You start with the angle measure in degrees. For example, in our problem, it's \(-320^{\circ}\).
• Apply the formula by replacing 'degrees' with \(-320\) which gives us: \[radians = -320 \times \frac{\pi}{180}\]
• Once you substitute the values, it boils down to a multiplication operation involving \(\pi\).

Using \(\pi\) in calculations ensures you get the exact radian measure in terms of \(\pi\). This is particularly useful because \(\pi\) is an infinite, non-repeating decimal, and using it symbolically maintains precision.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, especially in simplifying the conversion results. After substituting degrees into the conversion formula, you often end up with a fraction. To simplify, it's crucial to find the greatest common divisor (GCD) of the numerator and the denominator.For example, with our fraction \(-320 \times \frac{\pi}{180}\), we only need to simplify \(\frac{-320}{180}\).

• First, identify the GCD of 320 and 180, which is 20 in this case.
• Divide both the numerator and the denominator by their GCD: \[\frac{-320}{180} = \frac{-16}{9}\]
• This step reduces the fraction to its simplest form.

Finally, append \(\pi\) back into your simplified fraction to obtain the radian measure: \[\text{radians} = \frac{-16\pi}{9}\]Simplifying not only makes the fraction easier to work with but also ensures it is presented in a neat and comprehensible form.