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Find the reference angle for a rotation of
(10 pi)/(13).
Answer:

Find the reference angle for a rotation of $\frac{10 \pi}{13}$ . $\newline$ Answer: $\newline$

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Inverses of csc, sec, and cot

Full solution

Q. Find the reference angle for a rotation of

\frac{10 \pi}{13}

.

\newline

Answer:

\newline

Concept of Reference Angle: Understand the concept of a reference angle. A reference angle is the acute angle formed by the terminal side of an angle in standard position and the x-axis. It is always between $0$ and $\frac{\pi}{2}$ radians (or $0$ and $90$ degrees) and is positive.
Location in Unit Circle: Determine the location of the angle in the unit circle. $\newline$ Since $(10\pi)/(13)$ is less than $2\pi$ but greater than $\pi$ , the angle is located in the third quadrant of the unit circle.
Calculate Reference Angle: Calculate the reference angle. $\newline$ To find the reference angle for an angle in the third quadrant, subtract $\pi$ from the angle. $\newline$ Reference angle = $\frac{10\pi}{13} - \pi$
Perform Subtraction: Perform the subtraction to find the reference angle. $\newline$ Reference angle = $(\frac{10\pi}{13}) - (\frac{13\pi}{13})$ $\newline$ Reference angle = $(\frac{10\pi - 13\pi}{13})$ $\newline$ Reference angle = $-\frac{3\pi}{13}$ $\newline$ Since the reference angle must be positive, we take the absolute value. $\newline$ Reference angle = $|-\frac{3\pi}{13}| = \frac{3\pi}{13}$
Verify Correct Range: Verify that the reference angle is in the correct range. The reference angle of $\frac{3\pi}{13}$ is between $0$ and $\frac{\pi}{2}$ , which is the correct range for a reference angle.

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