Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\csc \theta \sec \theta \tan \theta$$

Step-by-Step Solution

Answer

The simplified expression is \( \sec^2 \theta \).

Step 1: Express all trigonometric functions in terms of sine and cosine

Begin by expressing each of the trigonometric functions \( \csc \theta \), \( \sec \theta \), and \( \tan \theta \) in terms of \( \sin \theta \) and \( \cos \theta \). \[ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta}. \] This yields the expression \[ \frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{\cos \theta}. \]

Step 2: Simplify the expression

Now simplify the expression by multiplying all terms: \[ \frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{\sin \theta}{\sin \theta \cos^2 \theta}. \] The \( \sin \theta \) in the numerator and denominator cancel out, leaving \[ \frac{1}{\cos^2 \theta}. \]

Step 3: Utilize identity to further simplify

Recognize that \( \frac{1}{\cos^2 \theta} \) is equivalent to the trigonometric identity \( \sec^2 \theta \). Thus, the simplified expression is \( \sec^2 \theta \).

Key Concepts

Sine and Cosine Functions

Trigonometric functions are fundamental in mathematics, particularly in the study of triangles and waves. At the core of these functions lie the sine and cosine functions. The sine function, denoted as \( \sin \theta \), is a ratio comparing the opposite side to the hypotenuse in a right-angled triangle. Conversely, the cosine function, known as \( \cos \theta \), compares the adjacent side to the hypotenuse. These relationships are pivotal because they allow us to express other trigonometric functions. For instance:

• The cosecant function (\( \csc \theta \)) is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \).
• The secant function (\( \sec \theta \)) is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• The tangent function (\( \tan \theta \)) is the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Understanding these basic functions and their reciprocal relationships is essential for simplifying complex trigonometric expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable involved, assuming the expression is defined. These identities play a significant role in simplifying and solving trigonometric expressions or equations.One crucial identity is the Pythagorean Identity:\[\sin^2 \theta + \cos^2 \theta = 1\]From which we derive other identities:

• \( \sec^2 \theta = 1 + \tan^2 \theta \)
• \( \csc^2 \theta = 1 + \cot^2 \theta \)

In the solution above, recognizing \( \frac{1}{\cos^2 \theta} \) as \( \sec^2 \theta \) is a perfect illustration of using these identities to simplify expressions. By transforming complex expressions into more recognizable terms, trigonometric identities become powerful tools for problem-solving.

Step-by-Step Solution

Understanding the process of simplification is crucial. Let's revisit the step-by-step method used in solving the given exercise to ensure clarity.**Step 1: Conversion to Sine and Cosine**
Transform the original expression \( \csc \theta \sec \theta \tan \theta \) into terms of sine and cosine using the definitions:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This results in \( \frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{\cos \theta} \).**Step 2: Simplification**
Multiply the terms:\[ \frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{\sin \theta}{\sin \theta \cos^2 \theta} \]Cancel \( \sin \theta \) from the numerator and denominator:\[ \frac{1}{\cos^2 \theta} \]**Step 3: Using Trigonometric Identity**
Recognize the result as a trigonometric identity, \( \sec^2 \theta \). Thus, simplifying \( \csc \theta \sec \theta \tan \theta \) requires changing functions, multiplying, and applying the identity for efficient results. This methodical approach showcases the importance of understanding each function and identity in trigonometry.