Problem 44 - 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.) $$\frac{\tan ^{2} \theta}{\sec ^{2} \theta}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite in Terms of Sine and Cosine

Begin by converting the trigonometric functions tangent and secant into terms of sine and cosine.Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).Substituting these into the expression gives:\[ \frac{\tan^2 \theta}{\sec^2 \theta} = \frac{(\frac{\sin \theta}{\cos \theta})^2}{(\frac{1}{\cos \theta})^2} = \frac{\frac{\sin^2 \theta}{\cos^2 \theta}}{\frac{1}{\cos^2 \theta}} \]

Step 2: Simplify the Expression

Now simplify the expression by multiplying the numerator by the reciprocal of the denominator:\[\frac{\sin^2 \theta}{\cos^2 \theta} \times \cos^2 \theta = \sin^2 \theta.\]The \( \cos^2 \theta \) terms in the numerator and denominator cancel out.

Key Concepts

Sine and Cosine

When working with trigonometric functions, sine and cosine are foundational elements that frequently appear. The sine function, denoted as \( \sin \theta \), represents the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse in a right triangle. On the other hand, the cosine function, denoted as \( \cos \theta \), is the ratio between the length of the adjacent side to the hypotenuse.
This makes sine and cosine essential in simplifying and solving trigonometric expressions. They set the groundwork for understanding more complex trigonometric functions, such as tangent and secant, which can be expressed in terms of sine and cosine.
Using these basic identities, many trigonometric expressions can be reduced to simpler forms, allowing for more straightforward calculations and understanding.

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is another primary trigonometric function. It represents the ratio of the sine of an angle to the cosine of the same angle. Essentially, it can be expressed by the equation \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This function is useful in various mathematical applications, including the simplification of complex expressions like the one in the exercise. By expressing tangent in terms of sine and cosine, calculations become more manageable. It helps break down difficult expressions into pieces using simpler trigonometric identities.
Understanding the tangent function's relation to sine and cosine is crucial for solving problems in trigonometry and further facilitating the application of trigonometric identities.

Secant Function

The secant function, expressed as \( \sec \theta \), is one of the reciprocal trigonometric functions. It is defined as the reciprocal of the cosine function, meaning it substitutes in as \( \sec \theta = \frac{1}{\cos \theta} \).
This function plays a significant role in trigonometry, particularly when working with angles in terms of their reciprocals. Using secant, complex trigonometric expressions can often be transformed into simpler forms. For example, dividing by \( \sec^2 \theta \) is equivalent to multiplying by \( \cos^2 \theta \), as shown in the provided solution.
Understanding the relationship between secant and cosine allows for easier manipulation of expressions, especially when aiming to simplify computations involving angles.

Trigonometric Identities

Trigonometric identities are equations that are true for any value of the variable within their domains. They are tools that offer shortcuts to simplify trigonometric expressions and solve equations. Some common identities include the reciprocal identities, such as \( \sec \theta = \frac{1}{\cos \theta} \), and Pythagorean identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \).
These identities serve as foundational blocks in converting and simplifying complex trigonometric expressions. For instance, by applying the identity for tangent in the original problem, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), the expression becomes more accessible.
Utilizing trigonometric identities is crucial for anyone studying trigonometry, as they allow for clearer analysis and solution of problems with reduced effort, turning cumbersome tasks into simpler operations.