Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Write expression as a single trigonometric function or a power of a trigonometric function. (You may wish to use a graph to support your result.) $$\cot \alpha \sin ^{2} \alpha \csc \alpha$$

Step-by-Step Solution

Answer

The expression simplifies to \(\cos \alpha\).

Step 1: Simplify the Expression Using Trigonometric Identities

To simplify the given expression \(\cot \alpha \sin ^{2} \alpha \csc \alpha\), we'll begin by breaking it down using trigonometric identities:\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha}, \quad \csc \alpha = \frac{1}{\sin \alpha} \].Substituting these identities into the expression:\[ \cot \alpha \sin ^{2} \alpha \csc \alpha = \left(\frac{\cos \alpha}{\sin \alpha}\right) \sin^{2} \alpha \left(\frac{1}{\sin \alpha}\right).\]

Step 2: Simplify Further by Canceling Terms

Next, perform the multiplication and cancel the common terms. Begin by multiplying what can be multiplied together:\[ \frac{\cos \alpha}{\sin \alpha} \cdot \sin^{2} \alpha \cdot \frac{1}{\sin \alpha} = \frac{\cos \alpha \cdot \sin^{2} \alpha}{\sin \alpha \cdot \sin \alpha}.\]Simplifying the fraction, notice that the \(\sin^{2} \alpha\) in the numerator and one \(\sin \alpha\) in the denominator cancel out:\[ \frac{\cos \alpha}{\sin \alpha} \cdot \sin \alpha = \cos \alpha.\]

Step 3: Write the Final Answer

The simplified form of the given expression \( \cot \alpha \sin ^{2} \alpha \csc \alpha \) is therefore:\[ \cos \alpha. \]This is the single trigonometric function equivalent to the given expression.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is fundamental when working with trigonometric expressions. These identities are equations involving trigonometric functions that hold true for all values of the variable. They help us simplify complex expressions by substituting equivalent terms.
To simplify the expression \( \cot \alpha \sin^{2} \alpha \csc \alpha \), we need to use the basic trigonometric identities:

• \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \)
• \( \csc \alpha = \frac{1}{\sin \alpha} \)

Substituting these identities transforms the original expression into a simpler form, making it easier to work with. These identities reduce clutter by breaking down the expression into terms with common trigonometric ratios, allowing for efficient simplification.

Simplifying Expressions

The process of simplifying expressions involves performing operations that reduce them to their simplest form. For trigonometric expressions, this often means applying identities, cancelling terms, and rearranging.
In our given expression \( \cot \alpha \sin^{2} \alpha \csc \alpha \), after substituting the trigonometric identities, the next step is to identify and cancel common terms. The expression becomes:
\[ \frac{\cos \alpha}{\sin \alpha} \cdot \sin^{2} \alpha \cdot \frac{1}{\sin \alpha} = \frac{\cos \alpha \cdot \sin^{2} \alpha}{\sin \alpha \cdot \sin \alpha} \]
Here, you recognize the \( \sin^{2} \alpha \) on top and bottom, allowing you to cancel out, leaving:
\[ \frac{\cos \alpha \cdot \sin \alpha}{\sin \alpha} \]
The remaining \( \sin \alpha \) in the numerator and denominator cancel, leading to the simplest form:
\[ \cos \alpha \]
Such simplifications are key in solving trigonometric problems efficiently, minimizing complexity by reducing expressions to their simplest, and often more recognizable form.

Trigonometric Functions

Trigonometric functions like sine, cosine, tangent, and their reciprocals (like cosecant \( \csc \), secant \( \sec \), and cotangent \( \cot \)) are the building blocks of trigonometry. Each of these functions relates an angle of a right-angled triangle to the ratios of two sides.
The exercise focuses on converting a multilayered expression into a single trigonometric function. Via simplification, \( \cot \alpha \sin^{2} \alpha \csc \alpha \) ultimately became \( \cos \alpha \). Achieving this required understanding that behind these operations are the fundamental characteristics of each trigonometric function:

• \( \cot \alpha \) which is the cosine of the angle divided by its sine.
• \( \csc \alpha \) is simply the inverse of sine.

Through these relationships, we can manipulate angles and trigonometric identities to arrive at expressions that are much easier to handle in both theoretical and practical applications. Such skills are useful in various fields, including physics, engineering, and even computer graphics.