Problem 18 - A Graphical Approach to Precalculus with Limits

Question

For e expression in Column I, choose the expression from Column II that completes an identity. You may have to rewrite one or both expressions. Do not use a calculator. \(\mathbf{I}\) \(-\tan x \cos x=\) _______ \(\mathbf{II}\) A. \(\frac{\sin ^{2} x}{\cos ^{2} x}\) B. \(\frac{1}{\sec ^{2} x}\) C. \(\sin (-x)\) D. \(\csc ^{2} x-\cot ^{2} x+\sin ^{2} x\) E. \(\tan x\)

Step-by-Step Solution

Answer

Expression from Column I is \(-\tan x \cos x = \sin(-x)\) from Column II (Option C).

Step 1: Rewrite the expression from Column I

Start by rewriting the given expression \(-\tan x \cos x\) from Column I. We know that \(\tan x = \frac{\sin x}{\cos x}\). Thus, we can rewrite the expression as \(-\frac{\sin x}{\cos x} \cdot \cos x\).

Step 2: Simplify the expression

Simplify the expression from Step 1: \(-\frac{\sin x}{\cos x} \cdot \cos x = -\sin x\). This is because the \(\cos x\) in the numerator and the \(\cos x\) in the denominator cancel each other out, leaving \(-\sin x\).

Step 3: Match with expressions from Column II

Now, compare the simplified expression \(-\sin x\) with each option given in Column II:- A. \(\frac{\sin ^{2} x}{\cos ^{2} x}\) - does not match \(-\sin x\).- B. \(\frac{1}{\sec ^{2} x}\) - not equivalent to \(-\sin x\).- C. \(\sin (-x)\) - this is equal to \(-\sin x\) because sine is an odd function.- D. \(\csc ^{2} x-\cot ^{2} x+\sin ^{2} x\) - not equivalent to \(-\sin x\).- E. \(\tan x\) - does not match \(-\sin x\).The expression that matches \(-\sin x\) is C. \(\sin (-x)\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental component of trigonometry that describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. It is an even function with a periodicity of \( 2\pi \), meaning it repeats itself every \( 2\pi \) radians. This function takes an input, typically an angle, and returns a value between -1 and 1.

This function is crucial in determining positions on the unit circle. Here, each angle corresponds to a coordinate on the circle's circumference, where the sine value represents the vertical component, or the \( y \)-coordinate. Because the circle is symmetrical, this gives sine an important property - it's an odd function. This means \( \sin(-x) = -\sin(x) \), which is exactly the reason why \( \sin(-x) \) equated to \(-\sin x\) in the solution exercise above.

When solving trigonometric equations, it's essential to consider these properties to simplify and identify equivalent expressions properly. Whether it's in transformations or solving identities, the behavior of the sine function guides the process.

Cosine Function

The cosine function, represented as \( \cos(x) \), is another essential trigonometric function, closely related to the sine function. It describes the ratio of the adjacent side to the hypotenuse in a right triangle. Like the sine function, cosine is periodic, repeating every \( 2\pi \). Its output values also range between -1 and 1.

The cosine function is particularly known for its role in defining the \( x \)-coordinate on the unit circle, representing the horizontal component of the angle's corresponding point. Unlike sine, cosine is an even function, which simply means \( \cos(-x) = \cos(x) \).

This relationship is useful when dealing with expressions and identities. You might see that in differential calculus, both sine and cosine have specific derivatives, with \( \frac{d}{dx} \cos(x) = -\sin(x) \). However, in our exercise, cosine function properties were used to simplify the initial expression \(-\tan x \cos x \) by cancelling \(\cos x\) in the numerator and denominator. This underscores the importance of understanding how cosine can affect other trigonometric operations.

Tangent Function

The tangent function, \( \tan(x) \), is a derivative relation entwined with sine and cosine. It is defined mathematically as \( \frac{\sin(x)}{\cos(x)} \). This relation makes the tangent inherently connected to both sine and cosine functions, as seen in our exercise.

Graphs of the tangent function visually represent periodicity with a period of \( \pi \). Unlike sine and cosine, tangent function values can vary from \(-\infty\) to \(\infty\), exhibiting asymptotic behavior where the function is undefined when \( \cos(x) = 0 \). This particularity means it breaks into vertical asymptotes, which reflects its periodicity.

Tangent also carries trigonometric identity relevance, particularly as it can be expressed in multiple forms, e.g., \( \tan(x) = \frac{1}{\cot(x)} \). Within the exercise, using the definition of tangent simplified \(-\tan(x) \cos(x)\) to \(-\sin(x)\), resonating again with the sine function's properties. Here, understanding tangent's interrelation unveils deeper links across trigonometric identities.