QuestionFind the reference angle for $\theta = -\frac{26 \pi}{7}$ . What is the least nonnegative coterminal angle $\theta_{C}$ ? In which quadrant is $\theta_{C}$ ?

Studdy Solution

STEP 1

Assumptions1. The given angle is $\theta=-\frac{26 \pi}{7}$ . The reference angle is the acute angle that the terminal side of the angle makes with the x-axis3. The angle of least nonnegative measure coterminal with $\theta$ is $\theta_{C}=\frac{ \pi}{7}$
4. The terminal side of $\theta_{C}$ lies in a certain quadrant, which we need to determine

STEP 2

First, we need to find the reference angle for $\theta=-\frac{26 \pi}{7}$ . The reference angle is always positive and less than or equal to $\frac{\pi}{2}$ . We can find it by taking the absolute value of $\theta$ and then subtracting the largest multiple of $\pi$ that is less than or equal to the absolute value.
$\theta_{R} = |\theta| - n\pi$ where $n$ is the largest integer such that $n\pi \leq |\theta|$ .

STEP 3

Now, plug in the given value for $\theta$ to calculate the reference angle.
$\theta_{R} = |-\frac{26 \pi}{7}| - n\pi$

STEP 4

Calculate the absolute value of $\theta$ .
$\theta_{R} = \frac{26 \pi}{7} - n\pi$

STEP 5

We can see that $n=3$ is the largest integer such that $n\pi \leq \frac{26 \pi}{7}$ , so we substitute $n=3$ into the equation.
$\theta_{R} = \frac{26 \pi}{7} -3\pi$

STEP 6

Calculate the reference angle.
$\theta_{R} = \frac{26 \pi}{} -3\pi = \frac{5 \pi}{}$

STEP 7

Now, we need to find the angle of least nonnegative measure coterminal with $\theta$ . This is the smallest positive angle that has the same terminal side as $\theta$ . We can find it by adding multiples of $2\pi$ to $\theta$ until we get a positive angle.
$\theta_{C} = \theta +2n\pi$ where $n$ is the smallest integer such that $\theta_{C} >0$ .

STEP 8

We are given that $\theta_{C}=\frac{2 \pi}{7}$ , so we don't need to calculate it.

STEP 9

Finally, we need to determine in what quadrant the terminal side of $\theta_{C}$ lies. We know that $\theta_{C}=\frac{2 \pi}{7}$ , which is less than $\frac{\pi}{2}$ , so the terminal side of $\theta_{C}$ lies in the first quadrant.
The reference angle for $\theta=-\frac{26 \pi}{7}$ is $\frac{5 \pi}{7}$ , the angle of least nonnegative measure coterminal with $\theta$ is $\frac{2 \pi}{7}$ , and the terminal side of $\theta_{C}$ lies in the first quadrant.

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