Problem 40 - Precalculus Mathematics for Calculus

Question

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=\frac{4 \pi}{3}$$

Step-by-Step Solution

Answer

Reference angle is \(\frac{\pi}{3}\); terminal point is \((-\frac{1}{2}, -\frac{\sqrt{3}}{2})\).

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 system. One complete revolution around the circle corresponds to an angle of \(2\pi\) radians.

Step 2: Determine the Reference Angle

For an angle \(t\), the reference angle \(t_{ref}\) is the acute angle formed by \(t\) with the x-axis. Since \(\frac{4\pi}{3}\) radians is in the third quadrant (between \(\pi\) and \(\frac{3\pi}{2}\)), the reference angle can be found by subtracting \(\pi\) from \(\frac{4\pi}{3}\):\[t_{ref} = \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}.\]

Step 3: Identify the Terminal Point on the Unit Circle

The terminal point corresponding to \(t\) can be found using the reference angle and the appropriate signs for the coordinates in the third quadrant. For \(\frac{\pi}{3}\), the coordinates would usually be \((\frac{1}{2}, \frac{\sqrt{3}}{2})\); however, in the third quadrant, both \(x\) and \(y\) are negative:\[(\cos(\frac{4\pi}{3}), \sin(\frac{4\pi}{3})) = (-\frac{1}{2}, -\frac{\sqrt{3}}{2}).\]

Key Concepts

Reference Angle

The reference angle is a crucial concept when working with angles on the unit circle. It helps simplify problems involving rotation and makes understanding trigonometric values easier. A reference angle is essentially the smallest positive angle that an angle of interest forms with the horizontal x-axis. For any angle in different quadrants, finding the reference angle allows us to relate it back to familiar values from the first quadrant.Given the angle \(t = \frac{4\pi}{3}\), which falls into the third quadrant (between \(\pi\) and \(\frac{3\pi}{2}\)), we determine the reference angle by calculating the absolute difference between \(t\) and \(\pi\) radians. Since \(\pi\) stands right across on the x-axis in standard position:

• The reference angle formula used is \(t_{ref} = t - \pi\).
• Inserting our angle: \(t_{ref} = \frac{4\pi}{3} - \pi = \frac{\pi}{3}\).

This result tells us that although \(\frac{4\pi}{3}\) is a more complex angle, it is closely linked to the simple acute angle \(\frac{\pi}{3}\). Reference angles thereby reveal a consistent pattern across the unit circle, where the trigonometric functions share equal magnitudes.

Radians

Radians are an important unit of angular measure used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians relate the angle to the radius directly. In the unit circle, the total angle around the circle is \(2\pi\) radians, equivalent to 360 degrees. Each radian essentially corresponds to the angle made when the arc length equals the radius of the circle.Understanding the conversion between degrees and radians can be helpful:

• The conversion formula is \(1\text{ radian} = \frac{180}{\pi}\) degrees.
• Therefore, \(\frac{4\pi}{3}\) radians can be converted to degrees by multiplying by \(\frac{180}{\pi}\).
• This results in \(\frac{4\pi}{3} \times 60 \approx 240\) degrees.

Choosing radian measure on a unit circle simplifies many mathematical operations, as the relationship with the circle's properties becomes more direct. Hence, radians are often used in higher-level mathematics and engineering.

Terminal Point

The terminal point is where a given angle lands on the unit circle. If you imagine placing the angle's vertex at the origin and tracing outwards from the positive x-axis, the terminal point is where you stop. For the angle \(t = \frac{4\pi}{3}\), determining this position accurately involves examining the angle's properties and which quadrant it lands in.To find the terminal point:

• Locate \(\frac{4\pi}{3}\) on the unit circle. The angle is in the third quadrant, meaning both x and y coordinates of the terminal point must be negative.
• The reference angle \(\frac{\pi}{3}\) is used to determine the point’s specifics. In the first quadrant, the coordinates for \(\frac{\pi}{3}\) would be \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
• Reflected into the third quadrant, these coordinates become \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).

In this way, terminal points show how angles spanning more than a full rotation map into positions on the circle, making it a powerful way to interpret angles larger than \(2\pi\) and relate angles between 0 and \(\pi\). This concept is pivotal in trigonometry, facilitating accurate graphing and function evaluation.