Question

sketch θ = \frac{5pi}{4} in standard position. drag to show the angle.

Explanation:

Step1: Recall standard - position angle concept

An angle $\theta$ in standard position has its vertex at the origin $(0,0)$ of the coordinate - plane and its initial side along the positive $x$ - axis.

Step2: Analyze the given angle $\theta=\frac{5\pi}{4}$

We know that $\pi$ radians is equivalent to $180^{\circ}$, so $\frac{5\pi}{4}=\frac{5\times180^{\circ}}{4}=225^{\circ}$.

Step3: Determine the terminal side

Since $180^{\circ}<225^{\circ}<270^{\circ}$, the terminal side of the angle $\theta = \frac{5\pi}{4}$ lies in the third quadrant. To sketch it, start from the positive $x$ - axis (initial side) and rotate counter - clockwise by an angle of $\frac{5\pi}{4}$ radians (or $225^{\circ}$).

Answer:

To sketch $\theta=\frac{5\pi}{4}$ in standard position, place the vertex at the origin, the initial side on the positive $x$ - axis, and rotate counter - clockwise by an angle of $\frac{5\pi}{4}$ radians. The terminal side will be in the third quadrant. (Since this is a sketching instruction, a verbal description is sufficient. If a visual sketch was required, one would draw a unit circle centered at the origin, mark the positive $x$ - axis as the initial side, and then draw a ray from the origin into the third quadrant making an angle of $\frac{5\pi}{4}$ radians with the positive $x$ - axis).