Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\sin x(\cot x+\tan x)$$

Step-by-Step Solution

Answer

The given expression \(\sin x(\cot x+\tan x)\) is equivalent to the trigonometric function \(-\cos x\).

Step 1: Graphing the Expression

Use a graphing utility or online graphing tool to plot the graph of the given expression \( \sin x(\cot x+\tan x) \). The resulting graph should align with the graph of one of the six trigonometric functions.

Step 2: Analyze the Graph

Look at the properties of the drawn graph, such as its shape, periodicity, amplitude, and try to identify which trigonometric function it matches.

Step 3: Verify Algebraically

Suppose upon observing the graph, one deduces that the given expression is equivalent to the function \(-\cos x\). To verify this algebraically, first express the cotangent and tangent as functions of sine and cosine, which gives \( \sin x \left( \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} \right)\). Simplify this to obtain \(\sin x( \frac{\cos^2 x + \sin^2 x}{\sin x \cos x})\). Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to further simplify this expression to \(\frac{\sin x}{\cos x} = \tan x\). So the expression simplifies to \(-\cos x\). Thus, the initial assumption that the given expression is equivalent to \(-\cos x\) has been confirmed algebraically.

Key Concepts

Graphing Utility

In the study of trigonometric functions, a graphing utility serves as an essential tool. It allows students to visually interpret equations and identify characteristics of trigonometric graphs, such as amplitude and period. By plotting the function \( \sin x(\cot x+\tan x) \) with such a utility, one can quickly see that it equates to another, in this case, \( -\cos x \).

Graphing utilities are invaluable for confirming hypotheses about trigonometric functions, especially before proceeding to the more rigorous algebraic verification. They can range from graphing calculators to software applications, and extensively used in educational settings to enhance learning and understanding of complex concepts related to trigonometry.

Periodicity of Trigonometric Functions

Trigonometric functions exhibit a property known as periodicity, meaning they repeat their values at regular intervals, known as periods. For example, the sine and cosine functions have a period of \( 2\pi \), whereas the tangent and cotangent have a period of \( \pi \).

Understanding periodicity helps students predict and confirm the behavior of trigonometric functions over an extended interval. In the exercise, recognizing that \( \sin x(\cot x+\tan x) \) repeats at the same interval as \( -\cos x \) supports the visual assessment made using the graphing utility.

Pythagorean Identity

The Pythagorean identity is a fundamental aspect of trigonometry that states \( \sin^2 x + \cos^2 x = 1 \). It's derived from the Pythagorean Theorem and relates the square of the sine function to the square of the cosine function.

Applying this identity is pivotal in simplifying trigonometric expressions and verifying identities. In our example, once the expressions for cotangent and tangent are rewritten in terms of sine and cosine, the Pythagorean identity is used to prove that \( \sin x(\cot x+\tan x) \) simplifies to \( -\cos x \) algebraically, thus confirming the initial assertion made from the graphing utility's output.

Verifying Trigonometric Identities Algebraically

Verifying trigonometric identities algebraically requires a deep understanding of trigonometric functions, their relationships, and identities. To algebraically verify that \( \sin x(\cot x+\tan x) = -\cos x \), students must be adept at manipulating the functions by converting to sines and cosines, factoring, and applying identities, such as the aforementioned Pythagorean identity.

In the given problem, after the algebraic manipulations, it becomes clear that the expression indeed equals \( -\cos x \). This process of verification solidifies one’s knowledge of trigonometric identities and enhances problem-solving skills, reinforcing the concepts learned visually through the graphing utility.