Question

sketch \\(\theta = \frac{3\pi}{4}\\) in standard position.

Explanation:

Identify the angle and its quadrant

$$ \theta = \frac{3\pi}{4} = 135^\circ $$

$$ 90^\circ < 135^\circ < 180^\circ \implies \text{Quadrant II} $$

Determine the terminal side position

$$ \theta = \pi - \frac{\pi}{4} $$

$$ \text{Terminal side bisects Quadrant II at } 135^\circ \text{ relative to the positive } x\text{-axis.} $$

Answer:

To sketch the angle \(\theta = \frac{3\pi}{4}\) in standard position:

1. Start the initial side along the positive \(x\)-axis.
2. Rotate counterclockwise through Quadrant I (\(\frac{\pi}{2}\) or \(90^\circ\)) and halfway into Quadrant II.
3. Draw the terminal side in Quadrant II, exactly along the diagonal line that bisects the quadrant (at \(135^\circ\)).