Problem 48 - Algebra and Trigonometry

Question

\(39-52=\) Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\) $$ t=\frac{17 \pi}{4} $$

Step-by-Step Solution

Answer

(a) Reference number: \( \frac{\pi}{4} \); (b) Terminal point: \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Step 1: Identify the reference number

The reference number for a given angle in radians is the equivalent angle within the range [0, 2π). Given that the angle is \( t = \frac{17\pi}{4} \), we first need to convert this angle into an equivalent angle between 0 and 2π by using modulo 2π. This involves finding the remainder when \( \frac{17\pi}{4} \) is divided by \( 2\pi \).

Step 2: Simplify the angle by reducing modulo 2π

First, express \( 2\pi \) in terms of \( \pi/4 \) to align the denominators: \( 2\pi = \frac{8\pi}{4} \). Now, divide \( \frac{17\pi}{4} \) by \( \frac{8\pi}{4} \) to see how many full circles it completes.Calculate the division:\( \frac{17/4}{8/4} = \frac{17}{8} = 2 \text{ R } 1 \). This means \( \frac{17\pi}{4} = 2 imes 2\pi + \frac{ ext{1} imes\pi}{4} \). Therefore, the angle equivalent to \( t = \frac{17\pi}{4} \) that is less than \( 2\pi \) is \( \frac{\pi}{4} \). Thus, the reference number is \( \frac{\pi}{4} \).

Step 3: Determine the terminal point

The terminal point is determined by the angle's position on the unit circle. Since the reference angle is \( \frac{\pi}{4} \), its corresponding point on the unit circle is in the first quadrant where both cosine and sine are positive.The coordinates for the terminal point at an angle of \( \frac{\pi}{4} \) is calculated using:- \( \cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).Therefore, the terminal point is \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Key Concepts

Reference Angle

The reference angle is an essential concept when working with trigonometric functions. It simplifies the process of finding angles by reducing any given angle to its equivalent within the first rotation, or the range of [0, 2\pi). This way, no matter how large or small the angle, you can find its "simplified" version. To find a reference angle, we typically convert an angle larger than 2\pi or less than 0 by subtracting or adding multiples of 2\pi, respectively, until it falls within the desired range.

For example, with an angle of \( \frac{17\pi}{4} \), we need to determine how many complete rotations of \( 2\pi \) that encompasses. By using modulo (remainder division), we reduce it to the reference angle which is \( \frac{\pi}{4} \). This means after several full circles, the angle reaches \( \frac{\pi}{4} \), which is its position in the first quadrant of the unit circle. Understanding this reduction is key to tackling many trigonometric problems.

Radians

Radians are a unit of angular measure used in many areas of mathematics, especially in trigonometry. Unlike degrees, which measure angles in portions of a "full rotation" (360 degrees), radians use the radius of a circle to define the angle.

A full circle in radians is \( 2\pi \). Thus, angles are often expressed in terms of \( \pi \) as seen in expressions like \( \frac{\pi}{4} \) or \( \frac{17\pi}{4} \). Radians provide a more natural measure for angles when dealing with mathematical concepts like the unit circle.

• Degrees: Out of 360 parts of a full circle.
• Radians: Out of \( 2\pi \), or the circle's circumference divided by its radius.

For practical conversion, remember: 180 degrees = \( \pi \) radians. Thus, \( 1 \) radian is approximately \( 57.3 \) degrees. Expressing angles in radians is crucial for understanding functions on the unit circle, and allows easier integration into calculus.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate system. Its primary usage is in defining trigonometric functions for all real angles.

On the unit circle, any angle \( t \) corresponds to a certain point \((x, y)\) on the circle, which can be expressed as \((\cos(t), \sin(t))\). Why is this important?

• It provides a way to define cosine and sine values for any angle through precise coordinates.
• It allows for easy visual understanding of angles and their corresponding values.

For instance, an angle of \( \frac{\pi}{4} \) on the unit circle leads to the terminal point \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\), as both cosine and sine are equal at this angle. This representation helps not only in visualizing angles but also in computing trigonometric functions for applications involving rotation, wave functions, and more.

Cosine and Sine

Cosine and sine are basic trigonometric functions directly derived from the unit circle. They define the relation of an angle to the coordinates on this circle.

When you have an angle \( t \),

• The cosine value, \( \cos(t) \), gives the x-coordinate of the angle’s terminal point on the unit circle.
• The sine value, \( \sin(t) \), provides the y-coordinate.

These functions oscillate between -1 and 1 and capture the circular motion effectively. For example, at an angle of \( \frac{\pi}{4} \), both the cosine and sine values are \( \frac{\sqrt{2}}{2} \), representing equal projection onto the x and y axes. Mastery of these functions is crucial for understanding wave behavior, harmonic motion, and periodic functions.

Terminal Point

The terminal point on the unit circle corresponds to the end position of an angle, measured in radians from the positive x-axis. Every angle points to a unique terminal point which can be described using cosine and sine values.

For an angle of \( \frac{\pi}{4} \), the terminal point is \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\). This point is found by projecting the angle onto the unit circle, ensuring it lies on the circle's circumference regardless of how big or small the angle is.

• First quadrant angles have both cosine and sine positive.
• Second quadrant has cosine negative, sine positive.
• Third quadrant makes both cosine and sine negative.
• Fourth quadrant has cosine positive, sine negative.

Understanding terminal points helps in visualizing rotations and oscillations in the context of trigonometric functions, forming the basis for complex number theory and vector analysis.