Question

Let \(f(x)=\sin x\) and \(g(x)=\cos x\) a) Write an expression as a quotient of functions that is equivalent to tan \(x .\) b) Write an expression as a product of functions that is equivalent to \(1-\cos ^{2} x\) c) Use graphing technology to verify your answers to parts a) and b).

Short answer

\(\tan x = \frac{f(x)}{g(x)}\), \(1 - \cos^2 x = f(x) \cdot f(x)\). Verify by plotting the functions.

Step-by-step solution

Understanding the Functions

Given two functions: \(f(x) = \sin x\) and \(g(x) = \cos x\).

Quotient of Functions for \(\tan x\)

Recall that \(\tan x = \frac{\sin x}{\cos x}\). Using the given functions, this can be written as: \(\tan x = \frac{f(x)}{g(x)}\).

Product of Functions for \(1 - \cos^2 x\)

Recall the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\). Rearrange it to get \(1 - \cos^2 x = \sin^2 x\). Thus, \(1 - \cos^2 x\) can be expressed as \((\sin x) \cdot (\sin x)\) or \(f(x) \cdot f(x)\).

Verifying Using Graphing Technology

Use a graphing calculator or software to plot \(\tan x\) and \(\frac{\sin x}{\cos x}\) on the same graph. They should overlap completely, verifying the expression. Similarly, plot \(\sin^2 x\) and \(1 - \cos^2 x\) to verify they overlap completely.

Key concepts

tan x

The tangent function, denoted as \(\tan x\), is one of the primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function. Simply put, if you know the values of \(\text{sin } x\) and \(\text{cos } x\), you can easily find \(\text{tan } x\). Mathematically, this relationship is expressed as: \[ \tan x = \frac{\sin x}{\cos x} \] This definition is helpful because it simplifies the process of finding the tangent value using known sine and cosine values. In the original exercise, by defining two functions: \(f(x) = \sin x\) and \(g(x) = \cos x\), the tangent function can be represented as: \[ \tan x = \frac{f(x)}{g(x)} \] This shows how we can express \(\tan x\) using the quotient of the functions \(\sin x\) and \(\cos x\).
Understanding \(\tan x\) is crucial because it appears frequently in various geometry and trigonometry problems.

Pythagorean identity

The Pythagorean identity is a fundamental trigonometric equation that relates the sine and cosine functions. It states that for any angle \(\theta\): \[ \sin^{2}\theta + \cos^{2}\theta = 1 \] This identity is derived from the Pythagorean theorem applied to the unit circle and is incredibly useful for simplifying and solving trigonometric equations. In the context of the given exercise, this identity is used to express the function \(1 - \cos^{2} x\). By rearranging the Pythagorean identity, we get: \[ \sin^{2}x + \cos^{2}x = 1 \] \[ 1 - \cos^{2}x = \sin^{2}x \] Thus, the expression \(1 - \cos^{2}x\) is equivalent to \(\sin^{2}x\).
This rearrangement shows the underlying relationship between sine and cosine and how squared trigonometric functions can be interconverted easily.

sin x

The sine function, denoted as \(\sin x\), is one of the primary functions in trigonometry. It represents the y-coordinate of a point on the unit circle corresponding to a given angle x measured in radians. The sine function oscillates between \( -1 \) and \( 1 \) and is periodic with a period of \( 2 \pi \). In simpler terms, it repeats its values every \( 2 \pi \) units. The formula for the sine function is: \[ \sin x = \frac{opposite}{hypotenuse} \] in the context of a right triangle. For the original exercise, defining \( f(x) = \sin x \) allows us to represent various trigonometric relationships using this function.

• For example, the sine function is used in defining the tangent function: \(\tan x = \frac{\sin x}{\cos x}\)
• It also appears in the expression derived from the Pythagorean identity: \(1 - \cos^{2} x = \sin^{2} x\)

Understanding the sine function helps in visualizing and solving many problems in trigonometry and periodic phenomena.

cos x

The cosine function, denoted as \(\cos x\), is another fundamental trigonometric function. It represents the x-coordinate of a point on the unit circle for a given angle x measured in radians. Similar to the sine function, the cosine function oscillates between \( -1 \) and \( 1 \) and is periodic with a period of \( 2 \pi \). The formula for the cosine function is: \[ \cos x = \frac{adjacent}{hypotenuse} \] in the context of a right triangle. In the given exercise, \( g(x) = \cos x \) is used to express tangent and other trigonometric identities.

• For instance, the function \(\tan x \) is derived using \( \frac{\sin x}{\cos x}\)
• The identity \( \cos^{2}x + \sin^{2}x = 1 \) helps in deriving expressions like \( 1 - \cos^{2}x = \sin^{2}x \)

Having a solid understanding of the cosine function is crucial because of its wide application in solving angular distance problems and other trigonometric analyses.