Problem 50 - Algebra and Trigonometry

Question

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{41 \pi}{4} $$

Step-by-Step Solution

Answer

Reference number: \( \frac{3\pi}{4} \); Terminal point: \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).

Step 1: Understand the Problem

We are given an angle \( t = -\frac{41\pi}{4} \) and are asked to find the reference number and the terminal point on the unit circle.

Step 2: Find the Coterminal Angle

We first need to convert \( t = -\frac{41\pi}{4} \) into an equivalent positive angle because angles in trigonometry are often considered modulo \( 2\pi \). We add \( 2\pi \) repeatedly to \( t \) until it becomes positive:\[ t' = -\frac{41\pi}{4} + 2\pi \times n = 2\pi k \ , \ ext{where} \ n \ ext{is an integer such that} \ t' > 0 \]Let's calculate it:\( 2\pi = \frac{8\pi}{4} \), hence to make \( -\frac{41\pi}{4} \) positive, we find \( n \) such that\( t' = -\frac{41\pi}{4} + \frac{8\pi}{4} \times 11 \geq 0 \).Thus, \( -\frac{41\pi}{4} + 22\pi = \frac{43\pi}{4} \) is the coterminal angle.

Step 3: Find the Reference Number

The reference number is the shortest angle to the x-axis from the terminal point. For \( t' = \frac{43\pi}{4} \), we find how many full \( 2\pi \) rotations this has, by reducing it within \( \left[0, 2\pi\right) \):Subtract \( 2\pi = \frac{8\pi}{4} \) repeatedly from \( \frac{43\pi}{4} \) until it is between \( 0 \) and \( 2\pi \):\[ \frac{43\pi}{4} - 5 \cdot \frac{8\pi}{4} = \frac{43\pi}{4} - \frac{40\pi}{4} = \frac{3\pi}{4}. \]So, the reference number is \( \frac{3\pi}{4} \).

Step 4: Determine the Terminal Point

The terminal point on the unit circle is found using the angle's terminal position from 0, utilizing cosine and sine:Using \( \theta = \frac{3\pi}{4} \) to find the terminal point:1. Cosine: \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)2. Sine: \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \)Thus, the terminal point is \(\left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right).\)

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on the coordinate plane. They may look different initially, but by adding or subtracting full rotations (multiples of \( 2\pi \)), these angles coincide with one another.
For example, the angle given in the problem, \( t = -\frac{41\pi}{4} \), is negative, but we need to find its positive coterminal angle. To do this:

• We add \( 2\pi \) (or \( \frac{8\pi}{4} \)) repeatedly until the result is positive.

As demonstrated in the solution, after several steps, we find that \( \frac{43\pi}{4} \) is a positive coterminal angle with equivalent positioning to \( -\frac{41\pi}{4} \) on the unit circle.
This process ensures that angles are expressed in a universally understood positive range, simplifying further trigonometric calculations.

Reference Number

The reference number is a concept that helps simplify trigonometric calculations by finding the shortest path an angle takes to reach the x-axis.
It is essentially a positive acute angle formed between the terminal side of an angle and the x-axis. This is particularly useful because trigonometric functions are easy to calculate for reference angles.
To find it:

• We reduce the angle to within the interval \( \left[0, 2\pi\right) \) by subtracting full \( 2\pi \) rotations, as shown with \( \frac{43\pi}{4} \) reducing to \( \frac{3\pi}{4} \).

The reference number simplifies the understanding of angles' properties by reducing them to a common acute measure.

Unit Circle

The unit circle is a fundamental tool in trigonometry to visualize and calculate trigonometric functions. It is a circle with a radius of 1, centered at the origin of the coordinate plane.
Most importantly, it allows angles to be represented in radians, with their terminal points easily calculable as coordinates \((x, y)\).

• In the exercise, the unit circle is used to find the terminal point corresponding to an angle \( \theta = \frac{3\pi}{4} \).

On the unit circle, every point \((x, y)\) portrays coordinates based on \((\cos(\theta), \sin(\theta))\), highlighting the relationship between angle measurements and their trigonometric results.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent link angles to their respective ratios of triangle sides or coordinates on the unit circle.
In this example, after finding \( \theta = \frac{3\pi}{4} \), we determine the sine and cosine values:

• \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \)
• \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \)

These calculations locate the terminal point on the unit circle as \( (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \).
Trigonometric functions extend beyond simple angle measures to describe real-world cyclical and periodic behaviors, making them crucial in numerous scientific and engineering applications.