Question $\newline$ Show Examples $\newline$ Find an angle in each quadrant with a common reference angle with $165^{\circ}$ , from $0^{\circ} \leq \theta<360^{\circ}$ $\newline$ Answer Attempt $1$ out of $2$ $\newline$ Quadrant I: $\newline$ Quadrant II: $\newline$ Quadrant III: $\newline$ Quadrant IV: $\newline$ Submit Answer

View step-by-step help

Home

Math Problems

Algebra 2

Quadrants

Full solution

Q. Question

\newline

Show Examples

\newline

Find an angle in each quadrant with a common reference angle with

165^{\circ}

, from

0^{\circ} \leq \theta<360^{\circ}

\newline

Answer Attempt

1

out of

2

\newline

Quadrant I:

\newline

Quadrant II:

\newline

Quadrant III:

\newline

Quadrant IV:

\newline

Submit Answer

Understand Reference Angle: First, we need to understand what a reference angle is. A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For an angle in standard position (initial side along the positive x-axis), the reference angle is always between $0^\circ$ and $90^\circ$ . The angle of $165^\circ$ is in Quadrant II, and its reference angle is the acute angle it forms with the x-axis. $\newline$ To find the reference angle for $165^\circ$ , we subtract it from $180^\circ$ (since it's in Quadrant II). $\newline$ Reference angle = $180^\circ - 165^\circ = 15^\circ$ .
Quadrant I: Quadrant I: In this quadrant, the angle is the same as the reference angle because angles in Quadrant I are between $0^\circ$ and $90^\circ$ . So, the angle in Quadrant I with a reference angle of $15^\circ$ is simply $15^\circ$ .
Quadrant II: Quadrant II: This is the quadrant where our original angle, $165°$ , is located. Since we are looking for angles with the same reference angle, $165°$ is already the angle in Quadrant II with a reference angle of $15°$ .
Quadrant III: Quadrant III: In this quadrant, angles are between $180°$ and $270°$ . To find an angle with a reference angle of $15°$ , we add $180°$ to the reference angle. $\newline$ Angle in Quadrant III = $180° + 15° = 195°$ .
Quadrant IV: Quadrant IV: In this quadrant, angles are between $270^\circ$ and $360^\circ$ . To find an angle with a reference angle of $15^\circ$ , we subtract the reference angle from $360^\circ$ . $\newline$ Angle in Quadrant IV = $360^\circ - 15^\circ = 345^\circ$ .