Problem 64 - Precalculus Mathematics for Calculus

Question

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sec ^{2} t \sin ^{2} t, \cos t ; \quad\) any quadrant

Step-by-Step Solution

Answer

\( \sec^2 t \sin^2 t = \frac{1}{\cos^2 t} - 1 \).

Step 1: Identify Relationship Between Trigonometric Functions

The first expression involves \( \sec^2 t \sin^2 t \). Recall that \( \sec t = \frac{1}{\cos t} \), which means \( \sec^2 t = \frac{1}{\cos^2 t} \). Thus, the expression \( \sec^2 t \sin^2 t \) can be rewritten using \( \cos t \) as \( \frac{\sin^2 t}{\cos^2 t} \).

Step 2: Use Pythagorean Identity

The Pythagorean identity is given by \( \sin^2 t + \cos^2 t = 1 \). Solving for \( \sin^2 t \), we have \( \sin^2 t = 1 - \cos^2 t \). This allows us to express \( \sin^2 t \) in terms of \( \cos t \).

Step 3: Substitute \( \sin^2 t \) in the Expression

Substitute \( \sin^2 t = 1 - \cos^2 t \) into the expression from Step 1 to get \( \sec^2 t \sin^2 t = \frac{1 - \cos^2 t}{\cos^2 t} \).

Step 4: Simplify the Expression

Simplify the expression \( \frac{1 - \cos^2 t}{\cos^2 t} \) by dividing the numerator and denominator by \( \cos^2 t \). This gives \( \frac{1}{\cos^2 t} - 1 \).

Step 5: Final Expression

Thus, the expression \( \sec^2 t \sin^2 t \) in terms of \( \cos t \) is \( \frac{1}{\cos^2 t} - 1 \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec t \), is one of the fundamental trigonometric functions. It is directly related to the cosine function. The relationship between them is described by the following:

• \( \sec t = \frac{1}{\cos t} \)

In simpler terms, the secant function is the reciprocal of the cosine function. This means if you know the value of \( \cos t \), you can easily find \( \sec t \) by taking the reciprocal.

Understanding the secant function is crucial because it often appears in trigonometric identities and expressions, playing a key role in converting and simplifying them. Furthermore, the squared secant function, \( \sec^2 t \), is particularly significant in many expressions and equations, just as evidenced in our exercise.

Cosine Function

The cosine function, \( \cos t \), is one of the six primary trigonometric functions. It represents the horizontal coordinate or the x-coordinate of a point on the unit circle.

• \( \cos t \) helps in defining the ratio of the adjacent side to the hypotenuse in a right triangle

In trigonometric terms, \( \cos t\) is used extensively in various formulas, equations, and trig identities. For instance, the secant function relies heavily on \( \cos t \).

In any trigonometric expression, knowing the value of the cosine function allows you to explore other interrelated trigonometric functions. This is why it's advantageous to express trigonometric identities or expressions in terms of \( \cos t \). Because of its central role in trigonometry, both the implications in identities and in practical problem solving are significant.

Pythagorean Identity

The Pythagorean identity is a fundamental rule in trigonometry, which ties together sine and cosine functions:

• \( \sin^2 t + \cos^2 t = 1 \)

This fundamental identity enables us to express one trigonometric function in terms of another. For example, if you need to express \( \sin^2 t \) in terms of \( \cos t \), you rearrange the identity to:

• \( \sin^2 t = 1 - \cos^2 t \)

This rearrangement proved useful in our exercise, allowing substitution in complex expressions.

Applying the Pythagorean identity is a common strategy when working with trigonometric expressions and simplifying them. Such identities are not just memorization tasks, but tools for deriving more straightforward and computationally efficient expressions.

Trigonometric Expressions

Trigonometric expressions involve combinations of trigonometric functions, like sine, cosine, and secant, along with algebraic operations. These expressions are fundamental throughout trigonometry and calculus.

In our exercise, we dealt with an expression \( \sec^2 t \sin^2 t \). By substituting known identities and simplifying, such expressions can be rewritten in more familiar or convenient forms, like \( \frac{1}{\cos^2 t} - 1 \).

• Recognizing patterns is crucial for simplifying these types of expressions.
• Understanding the relationships between different functions allows for substitution and simplification.

Such skills are not only academic exercises but also valuable in real-world applications where algebraic and trigonometric expressions frequently require simplification or transformation. This characteristic makes trigonometry especially powerful in various disciplines like physics and engineering.