Math / Trigonometry

QuestionFind the reference angle for $\frac{41 \pi}{12}$ .

Studdy Solution

STEP 1

1. The angle is given in radians.
2. Reference angles are always positive and are the smallest angle between the terminal side of the given angle and the x-axis.
3. A full circle in radians is $2\pi$ .

STEP 2

1. Convert the given angle to an equivalent angle within $0$ to $2\pi$ .
2. Determine the quadrant of the angle.
3. Calculate the reference angle based on the quadrant.

STEP 3

First, we need to find an equivalent angle within $0$ to $2\pi$ . Since $\frac{41\pi}{12}$ is greater than $2\pi$ , we subtract $2\pi$ until the angle is within the desired range.

STEP 4

Calculate $2\pi$ in terms of twelfths:
$2\pi = \frac{24\pi}{12}$
Subtract $2\pi$ from $\frac{41\pi}{12}$ :
$\frac{41\pi}{12} - \frac{24\pi}{12} = \frac{17\pi}{12}$

STEP 5

Determine the quadrant of $\frac{17\pi}{12}$ . Since $\frac{17\pi}{12}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.

STEP 6

In the third quadrant, the reference angle is calculated as the angle minus $\pi$ .
$\frac{17\pi}{12} - \pi = \frac{17\pi}{12} - \frac{12\pi}{12} = \frac{5\pi}{12}$
The reference angle for $\frac{41\pi}{12}$ is:
$\boxed{\frac{5\pi}{12}}$

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QuestionFind the reference angle for $\frac{41 \pi}{12}$ .

Studdy Solution

STEP 1

1. The angle is given in radians.
2. Reference angles are always positive and are the smallest angle between the terminal side of the given angle and the x-axis.
3. A full circle in radians is $2\pi$ .

STEP 2

1. Convert the given angle to an equivalent angle within $0$ to $2\pi$ .
2. Determine the quadrant of the angle.
3. Calculate the reference angle based on the quadrant.

STEP 3

First, we need to find an equivalent angle within $0$ to $2\pi$ . Since $\frac{41\pi}{12}$ is greater than $2\pi$ , we subtract $2\pi$ until the angle is within the desired range.

STEP 4

Calculate $2\pi$ in terms of twelfths:
$2\pi = \frac{24\pi}{12}$
Subtract $2\pi$ from $\frac{41\pi}{12}$ :
$\frac{41\pi}{12} - \frac{24\pi}{12} = \frac{17\pi}{12}$

STEP 5

Determine the quadrant of $\frac{17\pi}{12}$ . Since $\frac{17\pi}{12}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.

STEP 6

In the third quadrant, the reference angle is calculated as the angle minus $\pi$ .
$\frac{17\pi}{12} - \pi = \frac{17\pi}{12} - \frac{12\pi}{12} = \frac{5\pi}{12}$
The reference angle for $\frac{41\pi}{12}$ is:
$\boxed{\frac{5\pi}{12}}$

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QuestionFind the reference angle for 41π12\frac{41 \pi}{12}1241π​.

Studdy Solution

STEP 1

1. The angle is given in radians.2. Reference angles are always positive and are the smallest angle between the terminal side of the given angle and the x-axis.3. A full circle in radians is 2π2\pi2π.

STEP 2

1. Convert the given angle to an equivalent angle within 000 to 2π2\pi2π.2. Determine the quadrant of the angle.3. Calculate the reference angle based on the quadrant.

STEP 3

First, we need to find an equivalent angle within 000 to 2π2\pi2π. Since 41π12\frac{41\pi}{12}1241π​ is greater than 2π2\pi2π, we subtract 2π2\pi2π until the angle is within the desired range.

STEP 4

Calculate 2π2\pi2π in terms of twelfths:2π=24π12 2\pi = \frac{24\pi}{12} 2π=1224π​Subtract 2π2\pi2π from 41π12\frac{41\pi}{12}1241π​:41π12−24π12=17π12 \frac{41\pi}{12} - \frac{24\pi}{12} = \frac{17\pi}{12} 1241π​−1224π​=1217π​

STEP 5

Determine the quadrant of 17π12\frac{17\pi}{12}1217π​. Since 17π12\frac{17\pi}{12}1217π​ is greater than π\piπ but less than 3π2\frac{3\pi}{2}23π​, it lies in the third quadrant.

STEP 6

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QuestionFind the reference angle for $\frac{41 \pi}{12}$ .

1. The angle is given in radians.
2. Reference angles are always positive and are the smallest angle between the terminal side of the given angle and the x-axis.
3. A full circle in radians is $2\pi$ .

1. Convert the given angle to an equivalent angle within $0$ to $2\pi$ .
2. Determine the quadrant of the angle.
3. Calculate the reference angle based on the quadrant.

First, we need to find an equivalent angle within $0$ to $2\pi$ . Since $\frac{41\pi}{12}$ is greater than $2\pi$ , we subtract $2\pi$ until the angle is within the desired range.

Calculate $2\pi$ in terms of twelfths:
$2\pi = \frac{24\pi}{12}$
Subtract $2\pi$ from $\frac{41\pi}{12}$ :
$\frac{41\pi}{12} - \frac{24\pi}{12} = \frac{17\pi}{12}$

Determine the quadrant of $\frac{17\pi}{12}$ . Since $\frac{17\pi}{12}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.