Problem 34 - 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.) $$\frac{\csc \theta \sec \theta}{\cot \theta}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \sec^2 \theta \).

Step 1: Rewrite Trigonometric Functions in Terms of Sine and Cosine

First, rewrite each trigonometric function in the expression in terms of sine and cosine:- \( \csc \theta = \frac{1}{\sin \theta} \)- \( \sec \theta = \frac{1}{\cos \theta} \)- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)Thus, the expression \( \frac{\csc \theta \sec \theta}{\cot \theta} \) becomes \( \frac{\left(\frac{1}{\sin \theta}\right) \left(\frac{1}{\cos \theta}\right)}{\frac{\cos \theta}{\sin \theta}} \).

Step 2: Simplify the Expression

Now, simplify the expression by flipping the denominator and multiplying:\[ \frac{\frac{1}{\sin \theta \cos \theta}}{\frac{\cos \theta}{\sin \theta}} = \frac{1}{\sin \theta \cos \theta} \times \frac{\sin \theta}{\cos \theta} \]Simplify by canceling common factors:\[\frac{1}{\cos^2 \theta}\]This can be rewritten as \( \sec^2 \theta \).

Step 3: Express as a Power of a Single Trigonometric Function

Finally, notice that \( \frac{1}{\cos^2 \theta} \) is equivalent to \( \sec^2 \theta \). So, the expression simplifies to a single trigonometric function:\[ \sec^2 \theta \].

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the field of precalculus and are commonly used in various branches of mathematics and physics. There are six primary trigonometric functions:

• Sine (\(\sin \theta \)): Represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos \theta \)): Represents the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan \theta \)): Represents the ratio of the sine to the cosine.
• Cosecant (\(\csc \theta \)): The reciprocal of sine, or \(\frac{1}{\sin \theta}\).
• Secant (\(\sec \theta \)): The reciprocal of cosine, or \(\frac{1}{\cos \theta}\).
• Cotangent (\(\cot \theta \)): The reciprocal of tangent, or \(\frac{1}{\tan \theta}\).

Understanding these functions is crucial because they allow you to manipulate and simplify complex expressions in mathematics, especially in the context of the precalculus course. These functions are interconnected through fundamental identities and assist in describing oscillations, waves, and many other natural phenomena.

Simplifying Expressions

Simplifying expressions is a key skill in trigonometry and precalculus that involves reducing a complex expression to its simplest form. This process often involves using trigonometric identities to rewrite expressions in a more compact form.
To simplify trigonometric expressions, you often:

• Use identities to rewrite functions in terms of sine and cosine. This converts everything into a common base, making it easier to see relationships and cancel terms.
• Factor expressions or use reciprocal identities to eliminate complex fractions. For example, turning \(\csc \theta\) into \(\frac{1}{\sin \theta}\). This helps align terms for possible simplification.
• Cancel common factors when possible. As seen in the solution, flipping the denominator allows you to multiply and subsequently cancel terms to simplify.

Let's look at our example: from the original expression\(\frac{\csc \theta \sec \theta}{\cot \theta}\), you simplify by rewriting everything in terms of basic trigonometric functions. Then, you perform arithmetic operations to simplify it step by step, ultimately leading to \(\sec^2 \theta\), a simpler representation.

Precalculus

Precalculus serves as the foundational groundwork for calculus. It includes a study of functions, algebra, and trigonometry, and is designed to prepare students for the concepts of limits, derivatives, and integrals in calculus.
Trigonometric functions are crucial in precalculus because they provide the tools needed for solving equations involving angles. In a precalculus course:

• Students learn to solve trigonometric equations and apply identities to simplify expressions.
• They explore the unit circle and understand the periodic nature of trigonometric functions.
• They apply these skills to model real-life phenomena such as sound waves and circular motion.

Understanding how to manipulate and simplify trigonometric expressions—as demonstrated with the transformation from \(\frac{\csc \theta \sec \theta}{\cot \theta}\) to \(\sec^2 \theta\)—is vital for tackling more advanced problems in calculus. These foundational skills bridge the gap to calculus, where deeper analysis of changes and rates will build upon these trigonometric concepts.