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Problem 846

# What primary angle is coterminal with the angle of $$5(1 / 4) \pi$$ radians?

Expert verified
The primary angle coterminal with the angle of $5\frac{1}{4}\pi$ radians is $$\frac{-3}{4}\pi$$ radians.
See the step by step solution

## Step 1: Convert the mixed number to an improper fraction

First, we convert the mixed number $$5\frac{1}{4}$$ to an improper fraction: $5\frac{1}{4} = \frac{21}{4}$ Now, the given angle in radiants can be written as: $\frac{21}{4}\pi$

## Step 2: Subtract multiples of $$2\pi$$

Subtract multiples of $$2\pi$$ until we get an angle between $$0$$ and $$2\pi$$. We notice that $$8\pi$$ is the largest multiple of $$2\pi$$ that is less than $$\frac{21}{4}\pi$$. So, subtract $$8\pi$$ from our angle: $\alpha = \frac{21}{4}\pi - 8\pi = \frac{21}{4}\pi - \frac{32}{4}\pi =\frac{-11}{4}\pi$

## Step 3: Add 2$$\pi$$ if the result is negative

Since our result is negative, add $$2\pi$$ to the result to get the coterminal angle within the desired range: $\alpha + 2\pi = \frac{-11}{4}\pi + 2\pi = \frac{-11}{4}\pi + \frac{8}{4}\pi = \frac{-3}{4}\pi$ The primary angle coterminal with the angle of $$5\frac{1}{4}\pi$$ radians is $$\frac{-3}{4}\pi$$ radians.

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