Problem 43 - Algebra and Trigonometry

Question

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{13 \pi}{4} $$

Step-by-Step Solution

Answer

Reference angle: \( \frac{\pi}{4} \), Terminal point: \( \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \).

Step 1: Determine the Angle in Standard Position

To find the reference number and terminal point, first express \( t \) in terms of a full rotation. Since a full rotation is \( 2\pi \), calculate \( \frac{13\pi}{4} - 2k\pi \) where \( k \) is an integer. Divide \( \frac{13\pi}{4} \) by \( 2\pi \) to determine how many full rotations it has. \[ \frac{13\pi}{4} \div 2\pi = \frac{13}{8} \]. This equals \( 1.625 \), which means it has passed one full rotation, so we subtract \( 2\pi \) to find the equivalent angle in the first rotation: \( \frac{13\pi}{4} - 2\pi = \frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4} \).

Step 2: Find the Reference Angle

The reference angle for \( \theta = \frac{5\pi}{4} \) in standard position is the smallest angle between \( \theta \) and the x-axis. Since \( \frac{5\pi}{4} \) is in the third quadrant, the reference angle \( \theta_\text{ref} = \frac{5\pi}{4} - \pi = \frac{\pi}{4} \).

Step 3: Determine the Terminal Point

The terminal point determined by \( \theta = \frac{5\pi}{4} \) is the point \((x, y)\) on the unit circle that corresponds to this angle. Since \( \frac{5\pi}{4} \) is in the third quadrant, both x and y are negative. The calculations for \( x \) and \( y \) are as follows: \( x = \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( y = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). Therefore, the terminal point is \( \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \).

Key Concepts

Reference Angle

When thinking about angles, a reference angle can make understanding trigonometric functions much simpler. A reference angle is the smallest angle that a given angle makes with the x-axis. Its purpose is to help you determine the values of trigonometric functions for angles not in the first quadrant. The reference angle is always found within the first quadrant (0 to \( \frac{\pi}{2} \) radians or 0 to 90 degrees).To find the reference angle for any angle \( \theta \), you can follow these steps:

• If \( \theta \) is in the first quadrant, it's already the reference angle.
• In the second quadrant, subtract \( \theta \) from \( \pi \).
• In the third quadrant, subtract \( \theta \) by \( \pi \).
• And for those in the fourth quadrant, subtract \( \theta \) from \( 2\pi \).

For example, if \( \theta = \frac{5\pi}{4} \), it's located in the third quadrant. Thus, the reference angle is \( \frac{5\pi}{4} - \pi = \frac{\pi}{4} \). This is a handy angle since it's common and often used in trigonometric calculations.

Unit Circle

The unit circle is a fundamental concept in trigonometry and is essentially a circle with a radius of 1, centered at the origin of a coordinate plane. What makes the unit circle so useful is that it simplifies the use of sine and cosine, two key trigonometric functions.Whenever you're working with the unit circle:

• Each point on the circle's circumference can be described by the coordinates \((\cos \theta, \sin \theta)\).
• These coordinates correspond to the x and y positions on the plane for any angle \( \theta \).
• A full rotation (360 degrees or \( 2\pi \) radians) takes us completely around the circle, back to where we started.

For instance, the angle \( \theta = \frac{5\pi}{4} \) is on the unit circle. Its location tells us that both sine and cosine are negative in the third quadrant. Thus, the terminal point coordinates are \( (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) \), derived from \( \cos \) and \( \sin \) of the reference angle \( \frac{\pi}{4} \).

Terminal Point

A terminal point is where an angle lands on the unit circle. Picture an angle \( \theta \) rotating counterclockwise from the positive x-axis, stopping at a particular point on the circle's boundary.To determine a terminal point:

• First, identify the quadrant in which \( \theta \) resides after accounting for full rotations.
• Calculate the exact coordinates using cosine and sine functions.
• Remember, these functions give the x and y coordinates of the terminal point on the unit circle.

For example, to find the terminal point for \( \theta = \frac{5\pi}{4} \), observe that it falls in the third quadrant. Both the x and y values will be negative there, making them \(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\). Knowing the terminal points is essential in solving trigonometric equations, as they help visualize and predict the behaviour of sine and cosine values across different quadrants.