Problem 25 - Precalculus Mathematics for Calculus

Question

Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$t=\frac{5 \pi}{6}$$

Step-by-Step Solution

Answer

The terminal point is \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0) in the coordinate plane. Any angle "t" in the unit circle corresponds to an arc whose length is "t" and a point \(P(x, y)\) such that \(x^2 + y^2 = 1\,\channel{}{,}\).

Step 2: Determine the Reference Angle

The given angle \(t = \frac{5\pi}{6}\) is in radians. First, we find the reference angle. Since \(\frac{5\pi}{6} \) is in the second quadrant, the reference angle in the first quadrant is \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \).

Step 3: Calculate Cosine and Sine

Using the reference angle \( \frac{\pi}{6} \), we know: \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\) and \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\). Since \(\frac{5\pi}{6}\) is in the second quadrant, cosine is negative and sine is positive.

Step 4: Determine Terminal Point

The terminal point on the unit circle corresponding to \(t = \frac{5\pi}{6}\) is \(P(x, y) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\).

Key Concepts

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians offer a natural way to measure angles by relating them directly to the geometry of the circle. One full rotation around a circle is equivalent to an angle of 2π radians.

The radian measure of an angle is calculated by dividing the length of the arc by the radius of the circle. In the context of the unit circle—where the radius is 1—radian measurements offer a direct correlation to the arc they measure:

• 1 radian is approximately 57.2958 degrees.
• π radians equal 180 degrees, which makes π/6 radians 30 degrees.
• To convert radians to degrees, multiply by 180/π.

Understanding radians is crucial when working with trigonometric functions on the unit circle, as these functions often use radians to describe angles.

Reference Angle

A reference angle is the smallest angle that a given angle makes with the x-axis. Understanding reference angles is crucial in trigonometry, especially when dealing with angles in different quadrants.

To find the reference angle:

• For angles in the first quadrant, the reference angle is the angle itself.
• For those in the second quadrant, subtract the angle from π.
• In the third quadrant, subtract π from the angle.
• For the fourth quadrant, subtract the angle from 2π.

In the exercise where we have the angle \(rac{5\pi}{6}\), we determine that it is in the second quadrant. The reference angle is found by subtracting \(rac{5\pi}{6}\) from \(\pi\), giving \(rac{\pi}{6}\). Reference angles help to easily determine sine, cosine, and tangent values, as these trigonometric functions share signs with their reference angles.

Cosine and Sine Functions

The sine and cosine functions are vital components of the trigonometric world. They are periodic functions that help us understand the properties of triangles and circles.

Let's break down the sine and cosine functions and their behavior on the unit circle:

• The cosine of an angle in the unit circle is equivalent to the x-coordinate of the corresponding point on the circle.
• The sine of an angle corresponds to the y-coordinate.
• These functions are continuous and oscillate between -1 and 1.

Given a reference angle of \(\frac{\pi}{6}\), its cosine is \(\frac{\sqrt{3}}{2}\) and sine is \(\frac{1}{2}\). In the second quadrant where \(\frac{5\pi}{6}\) lies, cosine becomes negative, while sine remains positive, leading to the terminal point \(P(x, y) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\). Understanding these functions allows us to predict the behavior of angles and helps in analyzing waveforms, sound, and other oscillatory motions.