Problem 20 - Algebra and Trigonometry

Question

a. Use the unit circle shown for Exercises \(5-18\) to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part \((a)\) to find the value of the same trigonometric function at the indicated real number. $$ \begin{aligned} &a.\cos \frac{\pi}{3}\\\ &b.\cos \left(-\frac{\pi}{3}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The value of \(\cos(\pi/3)\) is 1/2 and the value of \(\cos(-\pi/3)\) is also 1/2.

Step 1: Use the unit circle to find the value of cosine at π/3

To use the unit circle to find the value of \(\cos(\pi/3)\), we need to remember that the \(x\)-coordinate of the point where the terminal side of the angle intersects the unit circle gives the cosine of the angle. With \(\pi/3\) radians or 60°, the \(x\)-coordinate is 1/2. So, \(\cos(\pi/3)=1/2\).

Step 2: Apply the properties of even and odd functions

Even functions are symmetric around the y-axis, which means \(f(-x) = f(x)\) for any input \(x\), while odd functions are symmetric around the origin, which means \(f(-x) = -f(x)\) for any input \(x\). For trigonometric functions, cosine is an even function. This means we can find \(\cos(-\pi/3)\) by taking the value of \(\cos(\pi/3)\) from Step 1 as the negative of the angle doesn't change the cosine value. Therefore, \(\cos(-\pi/3)=1/2\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to the ratios of sides of a right triangle. They are particularly useful in studying periodic phenomena. The core functions are sine, cosine, and tangent, each serving a unique purpose within trigonometry. Understanding these helps in analyzing waves, circular motion, and vibrations.

Here are the basics of the main trigonometric functions:

• Sine (\(\sin\)): This function relates to the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos\)): This function deals with the ratio of the adjacent side to the hypotenuse in a right triangle.
• Tangent (\(\tan\)): This function represents the ratio of the opposite side to the adjacent side.

These functions can be calculated using a unit circle, where the radius is 1. This simplifies the relationships because the hypotenuse, which is also the radius, remains constant. The angle's cosine corresponds to the x-coordinate of its termination on the unit circle, while the sine corresponds to the y-coordinate. This visual tool significantly aids in understanding trigonometric calculations.

Cosine Function

The cosine function, denoted as \(\cos(\theta)\), is vital in trigonometry. This function is even, meaning that it is symmetric with respect to the y-axis. It gives the horizontal position (x-coordinate) for a given angle on the unit circle.

The range of the cosine function spans from -1 to 1. For any angle \(\theta\), \(\cos(\theta)\) will yield a value within this range. In the context of the unit circle, the cosine of an angle directly gives the length along the x-axis where its terminal side intersects the circle.

This function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians. Here's a quick rundown of the key values within one period:

• At \(0\) or \(2\pi\): The cosine value is 1.
• At \(\pi/2\) or \(3\pi/2\): The cosine value is 0.
• At \(\pi\): The cosine value is -1.

These values form the peaks and transitions within the cosine wave. Calculating cosine for specific angles places them along these guidelines. Using the unit circle, like in the exercise, helps mainly to visualize these values and their applications.

Even and Odd Properties

In mathematics, the properties of being even or odd play a crucial role in simplifying expressions. A function is considered even if it satisfies \(f(-x) = f(x)\) for any choice of \(x\). Cosine is a prime example of an even function.

This means for any angle measured, both in positive and negative directions, the cosine produces the same result. For instance, if you find \(\cos(\pi/3) = 1/2\), then \(\cos(-\pi/3)\) will also be \(1/2\).

Odd functions, on the other hand, satisfy \(f(-x) = -f(x)\). Sine is an odd function. Understanding these properties helps solve equations more efficiently and reduces computational work.

• Even properties benefit: Quickly evaluate negative angle trigonometric functions.
• Odd properties benefit: Help in understanding phase shifts and symmetry in waves.

By using these properties, you avoid recalculating values unnecessarily, simplifying workloads especially when dealing with periodic functions.