Question

find the reference angle for the given angles. if the angle is given in degrees, give the reference angle in degrees. if it is given in radians, give the reference angle in radians. (a) $\frac{145pi}{26}$ part: 0 / 4 part 1 of 4 $\theta=$

Explanation:

Step1: Find the coterminal angle

First, we find a coterminal angle of $\frac{145\pi}{26}$ that is between $0$ and $2\pi$. We know that $2\pi=\frac{52\pi}{26}$. Divide $\frac{145\pi}{26}$ by $\frac{52\pi}{26}$: $\frac{145\pi}{26}\div\frac{52\pi}{26}=\frac{145}{52}=2\frac{41}{52}$. So, $\frac{145\pi}{26}=2\times2\pi+\frac{41\pi}{26}$.

Step2: Determine the quadrant

Since $\pi<\frac{41\pi}{26}<\frac{3\pi}{2}$ (because $\pi = \frac{26\pi}{26}$ and $\frac{3\pi}{2}=\frac{39\pi}{26}$), the angle $\frac{41\pi}{26}$ is in the third - quadrant.

Step3: Calculate the reference angle

For an angle $\theta$ in the third - quadrant, the reference angle $\theta'$ is given by $\theta - \pi$. So, $\theta'=\frac{41\pi}{26}-\pi=\frac{41\pi - 26\pi}{26}=\frac{15\pi}{26}$.

Answer:

$\frac{15\pi}{26}$