Question

trigonometry sketching an angle with absolute value less than 2π radians in standard... sketch θ = -\frac{5π}{6} in standard position.

Explanation:

Step1: Determine rotation direction

Negative angles rotate clockwise. So we'll rotate clockwise from the positive x - axis.

Step2: Find the measure of the angle in terms of standard positions

We know that $\pi$ radians is a straight angle (180 degrees) and $\frac{\pi}{6}$ radians is 30 degrees. The angle $\theta =-\frac{5\pi}{6}$ means we rotate clockwise $\frac{5\pi}{6}$ radians.
First, let's recall the unit circle divisions. A full circle is $2\pi$ radians, and it is divided into 12 equal parts (since $2\pi\div\frac{\pi}{6} = 12$) for angles with denominator 6.
Starting from the positive x - axis (rightmost point on the circle), rotating clockwise:

• Rotating $\frac{\pi}{6}$ (30 degrees) clockwise takes us to the direction of 330 degrees (or $-\frac{\pi}{6}$),
• Rotating $\frac{2\pi}{6}=\frac{\pi}{3}$ (60 degrees) clockwise takes us to 300 degrees (or $-\frac{\pi}{3}$),
• Rotating $\frac{3\pi}{6}=\frac{\pi}{2}$ (90 degrees) clockwise takes us to 270 degrees (or $-\frac{\pi}{2}$),
• Rotating $\frac{4\pi}{6}=\frac{2\pi}{3}$ (120 degrees) clockwise takes us to 240 degrees (or $-\frac{2\pi}{3}$),
• Rotating $\frac{5\pi}{6}$ (150 degrees) clockwise: since a full - rotation clockwise of $\pi$ radians (180 degrees) would take us to the negative x - axis, rotating $\frac{5\pi}{6}$ radians clockwise from the positive x - axis:

The positive x - axis is 0 radians. Moving clockwise, we can also think in terms of reference angles. The reference angle for $\theta =-\frac{5\pi}{6}$ is $\frac{5\pi}{6}$ (because the reference angle of a negative angle with magnitude less than $\pi$ is the angle itself in magnitude).
The angle $-\frac{5\pi}{6}$ is in the third quadrant? Wait, no. Wait, when we rotate clockwise from the positive x - axis:

• 0 to $-\frac{\pi}{2}$ (clockwise 90 degrees) is the fourth quadrant,
• $-\frac{\pi}{2}$ to $-\pi$ (clockwise 90 to 180 degrees) is the third quadrant.

Since $\frac{5\pi}{6}\approx150^{\circ}$, and $-\frac{5\pi}{6}$ is a clockwise rotation of 150 degrees. So from the positive x - axis, moving clockwise 150 degrees:
We know that 180 degrees clockwise is the negative x - axis. So 150 degrees clockwise is 30 degrees short of the negative x - axis (because 180 - 150 = 30). So the terminal side of the angle $\theta=-\frac{5\pi}{6}$ will be in the third quadrant, making a 30 - degree angle with the negative x - axis (or a 150 - degree angle with the positive x - axis in the clockwise direction).
On the given unit circle diagram, the marks on the circle: we can count the number of $\frac{\pi}{6}$ - radian steps. Since each step is $\frac{\pi}{6}$ radians.
Starting from the positive x - axis (rightmost point), moving clockwise:

• 1 step: $-\frac{\pi}{6}$ (330 degrees),
• 2 steps: $-\frac{2\pi}{6}=-\frac{\pi}{3}$ (300 degrees),
• 3 steps: $-\frac{3\pi}{6}=-\frac{\pi}{2}$ (270 degrees),
• 4 steps: $-\frac{4\pi}{6}=-\frac{2\pi}{3}$ (240 degrees),
• 5 steps: $-\frac{5\pi}{6}$ (210 degrees).

So we need to place the terminal side of the angle at the position that is 5 steps clockwise from the positive x - axis (each step is $\frac{\pi}{6}$ radians).

Answer:

To sketch $\theta =-\frac{5\pi}{6}$ in standard position:

1. Start at the positive x - axis (the right - most point on the unit circle).
2. Rotate clockwise (because the angle is negative) by $\frac{5\pi}{6}$ radians. This is equivalent to moving 5 "steps" of $\frac{\pi}{6}$ radians each in the clockwise direction around the unit circle. The terminal side of the angle will be in the third quadrant, making a reference angle of $\frac{\pi}{6}$ with the negative x - axis.