Problem 89 - Precalculus with Limits

Question

In Exercises 89 - 92, use a graphing utility to determine which of the six trigonometric functions is equal to the expression.Verify your answer algebraically. \( \cos x \cot x + \sin x \)

Step-by-Step Solution

Answer

The expression \(\cos x \cot x + \sin x\) is equal to \(1 - \sin^2 x + \sin x\).

Step 1: Rewrite the expression using trigonometric identities

Remember that the cotangent is the reciprocal of the tangent. So, you can rewrite \(\cot x\) as \(\frac{1}{\tan x}\). Now, replace \(\tan x\) with \(\frac{\sin x}{\cos x}\) which gives: \(\cos x . \frac{1}{\frac{\sin x}{\cos x}} + \sin x = \cos^2 x + \sin x\)

Step 2: Simplifying the equation

Use the Pythagorean identity, which states that \(\sin^2 x + \cos^2 x = 1\). Because \(\cos^2 x\) is present in the equation, replace \(\cos^2 x\) with \(1 - \sin^2 x\), giving you the expression: \(1 - \sin^2 x + \sin x\)

Step 3: Final Analysis and Graphing

Plot the function \(1 - \sin^2 x + \sin x\) using a graphing utility and confirm that it is equal to the original expression \(\cos x \cot x + \sin x\). Additionally, from the simplification in step 2, observe that this function has the form of a quadratic function \(y = a (u^2) + bu + c\), with \(u = \sin x\), \(a = -1\), \(b = 1\), and \(c = 1\).

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that relate the angles and sides of a triangle. They are used to simplify trigonometric expressions and solve trigonometric equations. One common identity is the reciprocal identity, such as \( \cot x = \frac{1}{\tan x} \). This identity shows the relationship between cotangent and tangent.
When simplifying trigonometric expressions, these identities are your best friends. They help you convert between different functions, such as expressing cotangent in terms of sine and cosine. For example, \( \cot x \) can be rewritten as \( \frac{\cos x}{\sin x} \).
This is useful for breaking down complex expressions into simpler parts, making them easier to work with. Regular practice with trigonometric identities sharpens problem-solving skills in trigonometry.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity reflecting the Pythagorean theorem in a trigonometric form. It states that \( \sin^2 x + \cos^2 x = 1 \). This identity is crucial in simplifying expressions like \( \cos^2 x + \sin x \).
In the given problem, we used the Pythagorean identity to replace \( \cos^2 x \) with \( 1 - \sin^2 x \). This substitution simplifies the expression to \( 1 - \sin^2 x + \sin x \).

• The Pythagorean identity helps verify the equality or simplify the expressions, ensuring accuracy in solutions.
• It transforms expressions by expressing one trigonometric function in terms of another.

Understanding this identity deeply aids in connecting various trigonometric functions and identities more easily.

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions. In this exercise, we use it to plot the function \( 1 - \sin^2 x + \sin x \) and verify its equivalence to the original expression.
Using a graphing utility allows you to observe the behavior of the function and check if two expressions yield the same graph. This method serves as a visual verification of algebraic manipulations.

• Graphing utilities facilitate exploring and understanding complex functions.
• They provide insight into changes in behavior with varied values, helping confirm theoretical solutions.

Thus, graphing ensures that algebraic simplifications are correct and gives learners confidence in their equation-solving abilities.