Problem 49 - 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.) $$\sin ^{2} \theta\left(\csc ^{2} \theta-1\right)$$

Step-by-Step Solution

Answer

\( \cos^2 \theta \)

Step 1: Express using basic trigonometric identities

The given expression is \( \sin^2 \theta (\csc^2 \theta - 1) \). We know \( \csc \theta = \frac{1}{\sin \theta} \). So, \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \). Substitute \( \csc^2 \theta \) with \( \frac{1}{\sin^2 \theta} \) in the original expression: \[ \sin^2 \theta \left( \frac{1}{\sin^2 \theta} - 1 \right) \]

Step 2: Simplify the expression inside the parenthesis

Subtract 1 from \( \frac{1}{\sin^2 \theta} \):\[ \frac{1}{\sin^2 \theta} - 1 = \frac{1 - \sin^2 \theta}{\sin^2 \theta} \]

Step 3: Simplify the entire expression

Substitute back into the expression:\[ \sin^2 \theta \cdot \frac{1 - \sin^2 \theta}{\sin^2 \theta} = 1 - \sin^2 \theta \]

Step 4: Use trigonometric identity to finalize

We know the identity \( 1 - \sin^2 \theta = \cos^2 \theta \). So, the simplified expression is \( \cos^2 \theta \).

Key Concepts

Sine and Cosine

In trigonometry, sine and cosine are foundational concepts used to describe aspects of right-angled triangles. These functions relate an angle of the triangle to the ratios of its sides. Specifically, for any angle \( \theta \) in a right triangle:

• The sine of an angle \( \theta \), denoted as \( \sin \theta \), is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle \( \theta \), represented as \( \cos \theta \), is the ratio of the length of the adjacent side to the hypotenuse.

In trigonometric identities, sine and cosine often work together. For example, the famous Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) allows us to express these functions in terms of each other. This identity highlights how these two functions are fundamental in simplifying trigonometric expressions. When faced with a trigonometric problem, transforming terms using sine and cosine can often make it more manageable.

Simplification

Simplification in mathematics involves transforming an expression into its most basic form. When dealing with trigonometric expressions, this often means using known identities to rewrite complex expressions in a simpler form.

For instance, take the expression \( \sin^2 \theta (\csc^2 \theta - 1) \). By converting terms into sine and cosine, as seen in the original solution, we simplified the parenthetical expression \( \frac{1}{\sin^2 \theta} - 1 \) to \( \frac{1 - \sin^2 \theta}{\sin^2 \theta} \).

• Distribute \( \sin^2 \theta \) across terms: \( \sin^2 \theta \cdot \frac{1 - \sin^2 \theta}{\sin^2 \theta} \) simplifies to \( 1 - \sin^2 \theta \).
• Apply the Pythagorean identity: Since \( 1 - \sin^2 \theta = \cos^2 \theta \), the expression further simplifies.

Simplification aids in providing clearer insight and solving more complex problems by making expressions more understandable.

Cosecant

Cosecant, abbreviated as \( \csc \), is one of the lesser-known trigonometric functions but plays an important role in certain mathematical contexts. It is defined as the reciprocal of the sine function. Thus, for an angle \( \theta \), the cosecant is given by:

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

When using the identity for cosecant, \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \), it allows us to rewrite expressions to make them easier to manage, like in the given problem.

Through this knowledge, you can manipulate terms from the known trigonometric identities to simplify expressions. In the original problem, by acknowledging that \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \), the expression becomes easier to work with and leads to the final simplified expression. Understanding these reciprocal functions deepens your overall grasp of trigonometry.