Problem 28 - Algebra and Trigonometry

Question

23-32 \(\approx\) Find the terminal point \(P(X, y)\) on the unit circle determined by the given value of \(t\) $$ t=\frac{5 \pi}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

To find the terminal point on the unit circle for a given angle \( t \), we use its coordinates \((X, Y)\), which are determined by \( \cos(t) \) and \( \sin(t) \) respectively for a given \( t \). Here, \( t = \frac{5\pi}{3} \).

Step 2: Determine the Reference Angle

Since \( t = \frac{5\pi}{3} \) is greater than \( 2\pi \), we need to find the equivalent angle between \( 0 \) and \( 2\pi \). Subtract \( 2\pi \) from \( \frac{5\pi}{3} \) to get the reference angle. \(\frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3}\). As this is negative, we can interpret it as the equivalent positive angle by adding \( 2\pi \) again, which yields \( \frac{\pi}{3} \).

Step 3: Identify the Quadrant

Since the reference angle \( \frac{\pi}{3} \) ends up in the first quadrant, and \( t \) itself revolves counter-clockwise past one full circle, it actually ends in the fourth quadrant for the position \( \frac{5\pi}{3} \).

Step 4: Determine Cosine and Sine Values

The known values for \( \frac{\pi}{3} \) in trigonometric functions indicate that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) and \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \). Since \( t = \frac{5\pi}{3} \) ends in the fourth quadrant, where cosine is positive and sine is negative, the coordinates are \( \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \).

Step 5: State the Terminal Point

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

Key Concepts

Terminal Point

The terminal point on the unit circle is a fundamental concept in trigonometry. It refers to the endpoint of an angle that begins at the origin and extends outward along the unit circle. Given an angle measurement, the terminal point is where this angle "lands" on the circle.

For the unit circle, which has a radius of 1, its coordinates are defined by the values of the trigonometric functions cosine and sine. The precise location of the terminal point is critical in identifying the values of these functions for the given angle.

Reference Angle

A reference angle is a useful concept in trigonometry, especially when working with angles greater than 360 degrees (or \[2\pi\] radians). It essentially helps simplify the calculation of trigonometric functions by reducing a large angle to a smaller, equivalent angle within the first quadrant.

To find the reference angle of \[t = \frac{5\pi}{3}\], we first see if the angle exceeds \[2\pi\]. If it does, subtract \[2\pi\] until it falls within one rotation of the circle. Here, subtracting gives us a positive reference angle \[\frac{\pi}{3}\]. This angle is used for quick reference to cos and sin values, which don't change with the angle's quadrant.

Trigonometric Functions

Trigonometric functions, namely sine and cosine, are pivotal to the study of the unit circle. For any angle \[t\] measured from the positive x-axis, cosine and sine provide the x and y coordinates, respectively, of the terminal point on the circle.

• Cosine: This defines the horizontal position, ranging from -1 to 1.
• Sine: This defines the vertical position, similarly ranging from -1 to 1.

For \[t = \frac{5\pi}{3}\], the cosine of \[\frac{\pi}{3}\] is \[\frac{1}{2}\], and the sine is \[\frac{\sqrt{3}}{2}\], indicating the exact coordinates of the unit circle path at this angle.

Quadrant

Quadrants help determine the sign of cosine and sine values. They divide the unit circle into four sections, each affecting whether these values are positive or negative.

• First Quadrant: Both cos and sin are positive.
• Second Quadrant: Cos is negative, sin is positive.
• Third Quadrant: Both are negative.
• Fourth Quadrant: Cos is positive, sin is negative.

For our angle \[t = \frac{5\pi}{3}\], which is located in the fourth quadrant, cosine is positive, and sine is negative. This informs us why the coordinates become \[\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\].

Cosine and Sine Values

Determining the cosine and sine values of an angle is integral for understanding its position on the unit circle. These values are not just numbers; they provide the x and y coordinates of an angle's terminal point.

For an angle \[\frac{\pi}{3}\], the standard trigonometric values are \[\cos(\frac{\pi}{3}) = \frac{1}{2}\] and \[\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\]. These provide the theoretical base for \[t = \frac{5\pi}{3}\], adjusted by its position in the fourth quadrant. Therefore, cos remains \[\frac{1}{2}\] while sin becomes \[-\frac{\sqrt{3}}{2}\], completing our understanding of the exact terminal point.