Math Problem Statement

What angle is at position (\(\frac{\sqrt{3}}{2}, -\frac{1}{2}\))?

Solution

The coordinates given in the problem are (32,12)\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right). These coordinates correspond to a point on the unit circle.

To determine the angle, we note that:

  • 32\frac{\sqrt{3}}{2} is the xx-coordinate.
  • 12-\frac{1}{2} is the yy-coordinate.

This point is in the fourth quadrant of the unit circle, where the cosine value (related to the xx-coordinate) is positive, and the sine value (related to the yy-coordinate) is negative.

In the first quadrant, the point (32,12)\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) corresponds to an angle of 3030^\circ or π6\frac{\pi}{6}. Since we are in the fourth quadrant, the angle is 36030=330360^\circ - 30^\circ = 330^\circ (or 2ππ6=11π62\pi - \frac{\pi}{6} = \frac{11\pi}{6} radians).

Thus, the angle at the position (32,12)\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right) is 330° (or 11π6\frac{11\pi}{6}).

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you determine angles in other quadrants using the unit circle?
  2. What is the relationship between radians and degrees?
  3. How do you convert an angle from degrees to radians?
  4. What are the sine and cosine values for common angles like 3030^\circ, 4545^\circ, and 6060^\circ?
  5. How do you find the reference angle for a given angle on the unit circle?

Tip: Knowing key points on the unit circle makes it easier to find angles based on coordinates.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Unit Circle
Radians and Degrees

Formulas

\(\cos(\theta) = \frac{\sqrt{3}}{2}\)
\(\sin(\theta) = -\frac{1}{2}\)
Conversion between radians and degrees

Theorems

Unit Circle Theorem

Suitable Grade Level

Grades 9-12