Problem 16 - Precalculus: A Concise Course

Question

Find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\). $$ t=\frac{5 \pi}{3} $$

Step-by-Step Solution

Answer

The coordinates of the point \((x, y)\) on the unit circle that correspond to the real number \(t=\frac{5 \pi}{3}\) are \((1/2, -\sqrt{3}/2)\).

Step 1: Calculate Cosine Value

First, calculate the cosine of \(t\), which equals to \(\cos\left(\frac{5\pi}{3}\right)\). Now, note that \(\frac{5\pi}{3}\) is equivalent to \(60^\circ\), and the cosine of \(60^\circ\) is \(1/2\). Therefore, \(\cos\left(\frac{5\pi}{3}\right) = 1/2\).

Step 2: Calculate Sine Value

Next, calculate the sine of \(t\), which equals to \(\sin\left(\frac{5\pi}{3}\right)\). Now, note that \(\frac{5\pi}{3}\) is equivalent to \(60^\circ\), and the sine of \(60^\circ\) is \(\sqrt{3} / 2\). However, because we're in the fourth quadrant (since \( \frac{5\pi}{3} \) is more than \( \pi \) and less than \( 2\pi \)), it's negative. Therefore, \(\sin\left(\frac{5\pi}{3}\right) = -\sqrt{3} / 2\).

Step 3: Determine the Coordinates

The coordinates \((x, y)\) on the unit circle that corresponds to \(t\) is \((\cos(t), \sin(t))\), therefore, the required coordinates are \((1/2, -\sqrt{3}/2)\).

Key Concepts

Trigonometric Functions

Trigonometric functions allow us to explore the relationships between the angles of a triangle and its side lengths. Specifically, these functions are essential for understanding the positions on a unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate system. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), which help us find the coordinates of a point on this circle.

- **Sine (\(\sin\))**: This function calculates the vertical coordinate of a point on the unit circle. It is the length of the opposite side over the hypotenuse in a right triangle.
- **Cosine (\(\cos\))**: This function determines the horizontal coordinate of a point. It is the length of the adjacent side over the hypotenuse.
- **Tangent (\(\tan\))**: This function is the ratio of \(\sin\) to \(\cos\).When we know an angle, these functions give us crucial points on the unit circle that simplify the process of finding angle-related values. They are crucial for solving problems involving rotations and oscillations, which are common in various fields of science and engineering.

Radians to Degrees Conversion

Angles can be measured in two main ways: degrees and radians. Converting between these is a necessary skill in trigonometry. One complete circle is divided into 360 degrees or \(2\pi\) radians. Therefore, the conversion factor is simply \(180/\pi\).

To convert radians to degrees, multiply the radian measure by \(180/\pi\). For instance, converting \(t = \frac{5\pi}{3}\) from radians to degrees involves:

• Multiply \(\frac{5\pi}{3}\) by \(\frac{180}{\pi}\). This simplification results in \(\frac{5 \times 180}{3} = 300\) degrees.

Conversely, to convert degrees to radians, multiply the degree value by \(\pi/180\). These conversions are essential when navigating between mathematical calculations that often depend on a specific angle measurement system.

Quadrants in Trigonometry

The unit circle is divided into four sections called quadrants, based on the coordinate axes. Each quadrant has specific properties:

- **First Quadrant**: Both the sine and cosine values are positive, occurring from 0 to \(\pi/2\) radians (or 0 to 90 degrees).
- **Second Quadrant**: Here, sine remains positive, but cosine switches to negative, from \(\pi/2\) to \(\pi\) radians (or 90 to 180 degrees).
- **Third Quadrant**: Both sine and cosine are negative, spanning from \(\pi\) to \(3\pi/2\) radians (or 180 to 270 degrees).
- **Fourth Quadrant**: Cosine turns positive, while sine is negative, covering from \(3\pi/2\) to \(2\pi\) radians (or 270 to 360 degrees).Understanding these quadrants is vital because the sign change affects the trigonometric calculations, as seen when finding the coordinates \((1/2, -\sqrt{3}/2)\) for \(t = \frac{5\pi}{3}\). Recognizing the placement of angles ensures accurate trigonometric function results.