Problem 65 - Algebra 2

Question

An angle of \(120^{\circ}\) is in standard position. What are the coordinates of the point at which the terminal side intersects the unit circle? \(\begin{array}{llll}{\text { A. } \frac{1}{2}, \frac{\sqrt{3}}{2}} & {\text { B. }\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)} & {\text { C. }\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)} & {\text { D. }\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)}\end{array}\)

Step-by-Step Solution

Answer

The coordinates are \((-1/2, \sqrt{3}/2)\)

Step 1: Converting Degrees to Radians

Convert the given angle in degrees to radians. This can be done by multiplying the degrees by \(\pi/180\). So, \(120^{\circ} \times \(\pi/180\) = \(\frac{2{\pi}}{3}\)

Step 2: Finding the coordinates

The coordinates on the unit circle are given by (cos(\(\theta\), sin(\(\theta\))). So, we find the values of cos(\(\theta\)) and sin(\(\theta\)) at \(\frac{2{\pi}}{3}\) radians. For cos(\(\frac{2{\pi}}{3}\))=-1/2 and sin(\(\frac{2{\pi}}{3}\))=\(\sqrt{3}/2\).

Step 3: Writing the final answer

Hence, the coordinates of the point on the unit circle corresponding to an angle of \(120^{\circ}\) in standard position are \((-1/2, \sqrt{3}/2)\). So, the answer is D: \((-1/2, \sqrt{3}/2)\).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand the trigonometric functions and their relationships with angles. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane (0,0). The significance of the unit circle lies in its ability to represent angles and their respective trigonometric values in a simplified manner. When an angle is placed in the unit circle in standard position (where the vertex of the angle is at the origin and the initial side lies along the positive x-axis), the terminal side will intersect the circumference of the circle at a specific point.

• This point is given in the form of (cos(𝜃), sin(𝜃)), where 𝜃 is the angle in standard position.
• Because the radius is 1, these coordinates directly represent the cosine and sine values of the angle.

The unit circle allows us to easily identify special angles and their trigonometric values, which is extremely useful in solving various mathematical problems.

Exploring Radian Conversion

Converting angles from degrees to radians, and vice versa, is a crucial skill in trigonometry. Degrees and radians are two different units for measuring angles, with radians being the standard unit used in mathematics.

• To convert degrees to radians, multiply the angle in degrees by \( \pi / 180 \).
• To convert radians to degrees, multiply the angle in radians by \( 180 / \pi \).

For example, to convert 120 degrees to radians:\[120^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{2\pi}{3}\]Understanding radian conversion is vital for working with the unit circle, as trigonometric functions in calculus and higher mathematics often involve radians.

Calculating Cosine and Sine Values

Once we have the angle in radians, finding the cosine and sine values is the next step in solving trigonometric problems, especially those involving the unit circle.For any angle \( \theta \) in the unit circle, the coordinates of the corresponding point are given by \((\cos(\theta), \sin(\theta))\). In our example with an angle of \( \frac{2\pi}{3} \), we find:

• \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \)
• \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \)

These values come from the reference angles and symmetry properties of the unit circle. By understanding these properties, we can quickly determine trigonometric values for various angles. For angles in different quadrants, sign conventions change based on the axis intersections, which is crucial for determining the correct values of cosine and sine.