Problem 27 - Algebra and Trigonometry

Question

In Exercises \(25-32,\) the unit circle has been divided into eight equal arcs, corresponding to t-values of $$ 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \pi, \frac{5 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}, \text { and } 2 \pi $$ a. Use the \((x, y)\) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. $$ \begin{array}{c} {a.\cos \frac{\pi}{2}} \\ {b. \cos \frac{9 \pi}{2}} \end{array} $$

Step-by-Step Solution

Answer

a. \( \cos(\frac{\pi}{2}) = 0\), b. \( \cos( \frac{9\pi}{2}) = 0 \).

Step 1: Find the cosine value for t = π/2

The point on the unit circle corresponding to \( t = \frac{\pi}{2} \) is (0, 1). The cosine function returns the x-coordinate of the point on the unit circle, which in this case is 0. So, \( \cos \frac{\pi}{2} = 0 \).

Step 2: Understand the periodic property of cosine function

The cosine function is periodic with period \( 2\pi \). This means that for any angle \( t \), \( cos(t) = cos(t + 2\pi*n) \) where \( n \) is any integer.

Step 3: Apply the periodic property to find the cosine value for t = 9π/2

To find the value of \( cos( \frac{9\pi}{2} ) \), we need to find an equivalent angle within one period of the cosine function i.e., within the range 0 - \( 2\pi \). \( \frac{9\pi}{2} = \frac{4\pi + \pi}{2} = 2\pi + \frac{\pi}{2}\). Given the periodic nature of the cosine function, \( \cos(2\pi + \frac{\pi}{2}) = \cos(\frac{\pi}{2}) \). Therefore \( \cos( \frac{9\pi}{2}) = \cos(\frac{\pi}{2}) = 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in connecting angles with sides of triangles, and they have specific values based on the unit circle. The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate plane. Trigonometric functions include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. These functions help us explore relationships between angles and lengths efficiently.

When we talk about these functions, particularly for the cosine and sine, they refer to specific points on the unit circle. In this context, the x-coordinate gives the cosine of the angle, while the y-coordinate provides the sine. This is incredibly useful to remember:

• Cosine uses the x-coordinate.
• Sine uses the y-coordinate.

This coordination helps simplify trigonometric calculations for any angle. Trigonometric functions are crucial not only in mathematics but also in physics, engineering, and even in everyday calculations.

Cosine Function

The cosine function is one of the primary trigonometric functions and represents the x-coordinate of a point on the unit circle at a given angle. Since the unit circle has a radius of 1, the cosine of an angle resembles how far the point on the circle is horizontally from the y-axis.

Suppose you need to find the cosine of an angle given in radians. Begin by identifying the corresponding point on the unit circle. For example, for a quarter of a circle, or \( \frac{\pi}{2} \) radians, the point is (0,1).

This means:

• \( \cos(\frac{\pi}{2}) = 0 \)

The cosine function tells us how the angle projects on the x-axis. Understanding this relationship is invaluable as it allows easy interpretation of angle measures and their trigonometric functions.

Periodic Properties

Periodicity is a key feature of trigonometric functions, especially with sine and cosine. It means these functions repeat their values in a regular pattern throughout their domains. For cosine, the periodic property is that every complete 360-degree, or \( 2\pi \) radians, results in the same function values.

This can be mathematically expressed as:

• \( \cos(t) = \cos(t + 2\pi \cdot n) \)

Where \( n \) is any integer value. By leveraging this property, complex angle measures can be simplified within one period, making calculations easier.

For instance, finding \( \cos(\frac{9\pi}{2}) \) involves understanding that adding \( 2\pi \) doesn't change the cosine's original value. Simplify \( \frac{9\pi}{2} \) to \( 2\pi + \frac{\pi}{2} \), which shows that

• \( \cos(\frac{9\pi}{2}) = \cos(\frac{\pi}{2}) \)

This fundamental property helps you recalibrate angles exceeding the typical range to focus on their standard values.

Radian Measure

Radians provide a natural measure of angles based on the radius of the circle. Unlike degrees, which divide a circle into 360 parts, radians express how far around a circle an angle takes you, relative to the circle's radius.

One complete revolution around the circle corresponds to \( 2\pi \) radians. This natural system aligns closely with calculus and is often more convenient in advanced mathematics and applied fields.

Key conversions to remember include:

• \( \pi \) radians = 180 degrees
• \( \frac{\pi}{2} \) radians = 90 degrees
• \( 2\pi \) radians = 360 degrees

Using radians, you can easily apply periodic properties. When working with radians, it's often easier to interpret results by relating them to familiar fractions like \( \pi \), \( \frac{\pi}{2} \), and \( 2\pi \). Understanding radians helps effectively express and calculate periodic functions on the unit circle.