Problem 43 - 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.) $$\frac{\cot ^{2} \theta}{\csc ^{2} \theta}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite Cotangent and Cosecant in Terms of Sine and Cosine

Start by rewriting the cotangent and cosecant functions in terms of sine and cosine. The cotangent function is defined as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). The cosecant function is \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute Definitions into Original Expression

Substitute the expressions for \( \cot \theta \) and \( \csc \theta \) into the original equation:\[ \frac{\cot^2 \theta}{\csc^2 \theta} = \frac{\left( \frac{\cos \theta}{\sin \theta} \right)^2}{\left( \frac{1}{\sin \theta} \right)^2} \]

Step 3: Simplify the Expression

Simplify the expression by dealing with the fractions:\[ \frac{\left( \frac{\cos \theta}{\sin \theta} \right)^2}{\left( \frac{1}{\sin \theta} \right)^2} = \frac{\frac{\cos^2 \theta}{\sin^2 \theta}}{\frac{1}{\sin^2 \theta}} \]This is the same as multiplying by the reciprocal:\[ \frac{\cos^2 \theta}{\sin^2 \theta} \times \sin^2 \theta \]

Step 4: Further Simplify to Get the Final Form

The \( \sin^2 \theta \) terms cancel out:\[ \cos^2 \theta \]Thus, the expression simplifies to \( \cos^2 \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying trigonometric expressions. They allow us to express functions like tangent, cotangent, secant, and cosecant in terms of the basic sine and cosine functions. For this exercise, we focus on the identities

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

Understanding these identities is vital, as they form the foundation for rewriting complex expressions and simplifying them effectively. They also make calculations more intuitive, as working with sine and cosine often simplifies the algebraic manipulation of trigonometric expressions. Recognizing these identities from the start can save time and make solving trigonometric problems easier.

Sine and Cosine Functions

The sine and cosine functions are the building blocks of trigonometry. These functions are defined based on the unit circle and describe the relationship between angles and the coordinates of points on the circle.
For any angle \( \theta \), the sine function is represented as the y-coordinate (or opposite side in a right triangle), written as \( \sin \theta \), and the cosine function as the x-coordinate (or adjacent side), known as \( \cos \theta \).
When dealing with trigonometric simplification, expressing functions in terms of sine and cosine can lead to more straightforward arithmetic manipulation.
For instance, in our original problem, rewriting \( \cot^2 \theta \) and \( \csc^2 \theta \) using sine and cosine makes the division and multiplication easier to handle, as the sine terms simplify out, leaving us with \( \cos^2 \theta \). This approach highlights the power of converting expressions to fundamental sine and cosine terms.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, are fundamental in both triangle measurement and the calculation of periodic phenomena.
Each function has a unique role and behavior in mathematical expressions, but they all can be interconverted using identities.
For this exercise, we took advantage of the relationships between these functions. By transforming cotangent and cosecant into expressions of sine and cosine, we gained a clearer perspective of the equation we needed to simplify. Recognizing the roles and definitions of these functions not only aids in solving such mathematical exercises but also provides a deeper understanding of their applications in physics, engineering, and other fields where wave-like phenomena occur, further proving their versatility.