Math / Trigonometry

Question15c. Is $\sin \frac{7 \pi}{6}$ positive or negative? negative positive
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Studdy Solution

STEP 1

1. We are dealing with the sine function, which is periodic and defined for all real numbers.
2. The angle $\frac{7\pi}{6}$ is given in radians.
3. We need to determine the sign of the sine function for this specific angle.

STEP 2

1. Convert the angle to degrees (optional for understanding).
2. Determine the reference angle.
3. Identify the quadrant in which the angle lies.
4. Determine the sign of the sine function in that quadrant.

STEP 3

Convert the angle $\frac{7\pi}{6}$ to degrees (optional):
$\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ$
This step is optional and just helps to visualize the angle in degrees.

STEP 4

Determine the reference angle for $\frac{7\pi}{6}$ :
The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. For angles in the third quadrant, the reference angle is found by subtracting $\pi$ from the angle:
$\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}$

STEP 5

Identify the quadrant in which the angle $\frac{7\pi}{6}$ lies:
Since $\frac{7\pi}{6}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.

STEP 6

Determine the sign of the sine function in the third quadrant:
In the third quadrant, the sine function is negative because both the x and y coordinates are negative, and sine corresponds to the y-coordinate.
The sine of $\frac{7\pi}{6}$ is $\boxed{\text{negative}}$ .

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Question15c. Is $\sin \frac{7 \pi}{6}$ positive or negative? negative positive
Submit step View next step

Studdy Solution

STEP 1

1. We are dealing with the sine function, which is periodic and defined for all real numbers.
2. The angle $\frac{7\pi}{6}$ is given in radians.
3. We need to determine the sign of the sine function for this specific angle.

STEP 2

1. Convert the angle to degrees (optional for understanding).
2. Determine the reference angle.
3. Identify the quadrant in which the angle lies.
4. Determine the sign of the sine function in that quadrant.

STEP 3

Convert the angle $\frac{7\pi}{6}$ to degrees (optional):
$\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ$
This step is optional and just helps to visualize the angle in degrees.

STEP 4

STEP 5

Identify the quadrant in which the angle $\frac{7\pi}{6}$ lies:
Since $\frac{7\pi}{6}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.

STEP 6

Determine the sign of the sine function in the third quadrant:
In the third quadrant, the sine function is negative because both the x and y coordinates are negative, and sine corresponds to the y-coordinate.
The sine of $\frac{7\pi}{6}$ is $\boxed{\text{negative}}$ .

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Question15c. Is sin⁡7π6\sin \frac{7 \pi}{6}sin67π​ positive or negative? negative positiveSubmit step View next step

Studdy Solution

STEP 1

1. We are dealing with the sine function, which is periodic and defined for all real numbers.2. The angle 7π6\frac{7\pi}{6}67π​ is given in radians.3. We need to determine the sign of the sine function for this specific angle.

STEP 2

1. Convert the angle to degrees (optional for understanding).2. Determine the reference angle.3. Identify the quadrant in which the angle lies.4. Determine the sign of the sine function in that quadrant.

STEP 3

Convert the angle 7π6\frac{7\pi}{6}67π​ to degrees (optional):7π6×180∘π=210∘ \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ 67π​×π180∘​=210∘This step is optional and just helps to visualize the angle in degrees.

STEP 4

STEP 5

Identify the quadrant in which the angle 7π6\frac{7\pi}{6}67π​ lies:Since 7π6\frac{7\pi}{6}67π​ is greater than π\piπ but less than 3π2\frac{3\pi}{2}23π​, it lies in the third quadrant.

STEP 6

Determine the sign of the sine function in the third quadrant:In the third quadrant, the sine function is negative because both the x and y coordinates are negative, and sine corresponds to the y-coordinate. The sine of 7π6\frac{7\pi}{6}67π​ is negative\boxed{\text{negative}}negative​.

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Question15c. Is $\sin \frac{7 \pi}{6}$ positive or negative? negative positive
Submit step View next step

1. We are dealing with the sine function, which is periodic and defined for all real numbers.
2. The angle $\frac{7\pi}{6}$ is given in radians.
3. We need to determine the sign of the sine function for this specific angle.

1. Convert the angle to degrees (optional for understanding).
2. Determine the reference angle.
3. Identify the quadrant in which the angle lies.
4. Determine the sign of the sine function in that quadrant.

Convert the angle $\frac{7\pi}{6}$ to degrees (optional):
$\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ$
This step is optional and just helps to visualize the angle in degrees.

Identify the quadrant in which the angle $\frac{7\pi}{6}$ lies:
Since $\frac{7\pi}{6}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.

Determine the sign of the sine function in the third quadrant:
In the third quadrant, the sine function is negative because both the x and y coordinates are negative, and sine corresponds to the y-coordinate.
The sine of $\frac{7\pi}{6}$ is $\boxed{\text{negative}}$ .