Math / Trigonometry

QuestionSketch $\theta=-\frac{7 \pi}{6}$ in standard position.

Studdy Solution

STEP 1

What is this asking? Draw an angle that's negative seven-sixths pi radians. Watch out! Negative angles go clockwise, not counter-clockwise!

STEP 2

1. Convert to Degrees
2. Draw the Angle

STEP 3

We're going to convert from radians to degrees, because sometimes it's easier to visualize degrees.
We know that $\pi$ radians is the same as $180$ degrees.
This gives us the conversion factor: $\frac{180^{\circ}}{\pi \text{ radians}}$

STEP 4

Let's multiply our angle, $-\frac{7\pi}{6}$ radians, by this conversion factor: $-\frac{7\pi}{6} \cdot \frac{180^{\circ}}{\pi}$

STEP 5

We can divide to one the $\pi$ in the numerator and denominator, which gives us: $-\frac{7}{6} \cdot 180^{\circ}$ We can also divide $180$ by $6$ , which gives us $30$ .
So, we have: $-7 \cdot 30^{\circ}$ Multiplying gives us: $-210^{\circ}$ So, $-\frac{7\pi}{6}$ radians is the same as $-210^{\circ}$ .
Awesome!

STEP 6

Remember, angles in standard position always start at the positive $\text{x}$ -axis.

STEP 7

Since our angle is negative, we'll rotate clockwise.
Imagine sweeping out an angle of $210^{\circ}$ in the clockwise direction.

STEP 8

$210^{\circ}$ clockwise is $30^{\circ}$ past $180^{\circ}$ in the clockwise direction.
That puts our angle in the second quadrant, if we were thinking about positive angles.

STEP 9

The angle $-\frac{7\pi}{6}$ radians, or equivalently $-210^{\circ}$ , is drawn starting from the positive x-axis, rotating $210^{\circ}$ clockwise, ending in the second quadrant (if we were thinking about positive angles).

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