Problem 38 - Precalculus

Question

Convert from radians to degrees. $$-6 \pi$$

Step-by-Step Solution

Answer

-6\pi radians is -1080 degrees.

Step 1: Understand the Conversion Formula

To convert radians to degrees, we use the formula \( ext{degrees} = ext{radians} imes \left( \frac{180}{\pi} \right) \). This formula is derived from the fact that \( \pi \) radians is equal to 180 degrees.

Step 2: Substitute the Radian Value

Substitute \(-6\pi\) for the radians in the formula. This gives us \(-6\pi \times \left( \frac{180}{\pi} \right)\).

Step 3: Simplify the Expression

In the expression \(-6\pi \times \left( \frac{180}{\pi} \right)\), the \(\pi\) in the numerator and the denominator cancels out. This leaves us with \(-6 \times 180\).

Step 4: Perform the Multiplication

Calculate the multiplication: \(-6 \times 180 = -1080\).

Step 5: Write the Final Answer in Degrees

So, \(-6\pi\) radians is equivalent to \(-1080\) degrees.

Key Concepts

Conversion Formula

Converting from radians to degrees is a straightforward process once you grasp the conversion formula. The key formula to remember is:\[\text{degrees} = \text{radians} \times \left( \frac{180}{\pi} \right)\]This formula stems from the relationship that \(\pi\) radians is exactly equivalent to 180 degrees. This means that when you want to convert any angle from radians to degrees, you multiply the radians by the fraction \(\frac{180}{\pi}\). This fraction acts as a conversion factor, changing the unit from radians, which is often used in calculus and trigonometry, to degrees, which is more commonly used in everyday contexts. Keep this formula handy whenever you're dealing with angle measurements.

Simplification

After setting up the conversion formula, the next step is simplification. Suppose we have the expression \[-6\pi \times \left( \frac{180}{\pi} \right)\] At this point, notice that both the numerator and the denominator have \( \pi \). The beauty of algebra allows us to cancel these terms. Cancellation simplifies the expression significantly:- The \(\pi\) in the numerator cancels with the \(\pi\) in the denominator.- After \(\pi\) cancellation, we are left with \[-6 \times 180\]This simplification step reduces complexity and makes the multiplication that follows much easier. Always look for opportunities to simplify an expression before proceeding to solve it. Not only does it make calculations easier, but it also reduces potential errors.

Multiplication

Once the expression is simplified, you can perform the multiplication step. The problem boils down to calculating\[-6 \times 180\]Here's how you can tackle it:

• Start by multiplying 6 and 180. This gives you 1080.
• Since our radian value was negative, make sure to remember that the resulting number will also be negative.

Hence, the multiplication yields \[-1080\]. Remembering to keep track of signs during multiplication is crucial, as the direction of the angle (negative in this case) affects the final degree measurement. So, the radian value of \(-6\pi\) radians translates to \(-1080\) degrees as the final answer.