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QuestionFind the reference angle for θ=29π6\theta=\frac{29 \pi}{6} and the least nonnegative angle coterminal with it. What quadrant is it in?

Studdy Solution

STEP 1

Assumptions1. The given angle is θ=29π6\theta=\frac{29 \pi}{6} . The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.
3. The coterminal angle is the angle of least nonnegative measure that shares the same terminal side as the given angle.
4. The quadrant of an angle is determined by the sign of its sine and cosine values.

STEP 2

First, we need to find the reference angle for θ=29π6\theta=\frac{29 \pi}{6}. To do this, we need to reduce the angle to an equivalent angle between 00 and 2π2\pi.
θ′=θmod  2π\theta' = \theta \mod2\pi

STEP 3

Now, plug in the given value for θ\theta to calculate θ′\theta'.
θ′=29π6mod  2π\theta' = \frac{29 \pi}{6} \mod2\pi

STEP 4

Calculate θ′\theta'.
θ′=29π6mod  2π=π6\theta' = \frac{29 \pi}{6} \mod2\pi = \frac{ \pi}{6}

STEP 5

Now that we have θ′\theta', we can find the reference angle. If θ′\theta' is in the first or fourth quadrant, the reference angle is θ′\theta' itself. If θ′\theta' is in the second or third quadrant, the reference angle is π−θ′\pi - \theta'.
θR=π−θ′\theta_{R} = \pi - \theta'

STEP 6

Plug in the value for θ′\theta' to calculate the reference angle.
θR=π−5π6\theta_{R} = \pi - \frac{5 \pi}{6}

STEP 7

Calculate the reference angle.
θR=π−5π6=π6\theta_{R} = \pi - \frac{5 \pi}{6} = \frac{\pi}{6}

STEP 8

Now that we have the reference angle, we can find the coterminal angle of least nonnegative measure. This is the angle θC\theta_{C} such that 0≤θC<2π0 \leq \theta_{C} <2\pi and θC\theta_{C} is coterminal with θ\theta.
θC=θmod  2π\theta_{C} = \theta \mod2\pi

STEP 9

Plug in the value for θ\theta to calculate θC\theta_{C}.
θC=29π6mod  2π\theta_{C} = \frac{29 \pi}{6} \mod2\pi

STEP 10

Calculate θC\theta_{C}.
θC=29π6mod  2π=5π6\theta_{C} = \frac{29 \pi}{6} \mod2\pi = \frac{5 \pi}{6}

STEP 11

Now that we have θC\theta_{C}, we can determine the quadrant in which the terminal side of θC\theta_{C} lies. If θC\theta_{C} is between 00 and π\frac{\pi}{}, it is in the first quadrant. If θC\theta_{C} is between π\frac{\pi}{} and π\pi, it is in the second quadrant. If θC\theta_{C} is between π\pi and 3π\frac{3\pi}{}, it is in the third quadrant. If θC\theta_{C} is between 3π\frac{3\pi}{} and π\pi, it is in the fourth quadrant.
Since 5π6\frac{5 \pi}{6} is between π\frac{\pi}{} and π\pi, the terminal side of θC\theta_{C} lies in the second quadrant.
The reference angle for θ=29π6\theta=\frac{29 \pi}{6} is π6\frac{\pi}{6}. The angle of least nonnegative measure coterminal with θ\theta is 5π6\frac{5 \pi}{6}. The terminal side of θC\theta_{C} lies in the second quadrant.

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