Question
Sketch the angle then find its reference angle -170^•
Answers
GPT-4o mini
To find the reference angle for \(-170^\circ\), we follow these steps:
1. **Locate the Angle**: Since \(-170^\circ\) is a negative angle, it is measured clockwise from the positive x-axis.
2. **Find the Positive Equivalent**: To convert the negative angle to a positive angle, add \(360^\circ\):
\[
-170^\circ + 360^\circ = 190^\circ
\]
This means that \(-170^\circ\) is coterminal with \(190^\circ\).
3. **Determine the Quadrant**: The angle \(190^\circ\) lies in the third quadrant since \(180^\circ < 190^\circ < 270^\circ\).
4. **Calculate the Reference Angle**: The reference angle is the acute angle formed with the x-axis. In the third quadrant, the reference angle can be found by subtracting \(180^\circ\) from the angle:
\[
190^\circ - 180^\circ = 10^\circ
\]
The reference angle for \(-170^\circ\) is therefore \(10^\circ\).
### Sketch of the Angle:
1. Draw a standard coordinate system with axes labeled.
2. From the positive x-axis, move clockwise to \(-170^\circ\).
3. This position corresponds to an angle of \(190^\circ\) in the standard counterclockwise direction, situating the terminal side in the third quadrant.
4. Mark the reference angle at \(10^\circ\) from the negative x-axis back to the terminal side.
### Final Answer:
The reference angle for \(-170^\circ\) is \(10^\circ\).
1. **Locate the Angle**: Since \(-170^\circ\) is a negative angle, it is measured clockwise from the positive x-axis.
2. **Find the Positive Equivalent**: To convert the negative angle to a positive angle, add \(360^\circ\):
\[
-170^\circ + 360^\circ = 190^\circ
\]
This means that \(-170^\circ\) is coterminal with \(190^\circ\).
3. **Determine the Quadrant**: The angle \(190^\circ\) lies in the third quadrant since \(180^\circ < 190^\circ < 270^\circ\).
4. **Calculate the Reference Angle**: The reference angle is the acute angle formed with the x-axis. In the third quadrant, the reference angle can be found by subtracting \(180^\circ\) from the angle:
\[
190^\circ - 180^\circ = 10^\circ
\]
The reference angle for \(-170^\circ\) is therefore \(10^\circ\).
### Sketch of the Angle:
1. Draw a standard coordinate system with axes labeled.
2. From the positive x-axis, move clockwise to \(-170^\circ\).
3. This position corresponds to an angle of \(190^\circ\) in the standard counterclockwise direction, situating the terminal side in the third quadrant.
4. Mark the reference angle at \(10^\circ\) from the negative x-axis back to the terminal side.
### Final Answer:
The reference angle for \(-170^\circ\) is \(10^\circ\).