Problem 50 - Algebra and Trigonometry

Question

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\cos t \sec t, \quad\) any quadrant

Step-by-Step Solution

Answer

The sign of \( \cos t \sec t \) is always positive.

Step 1: Express the Trigonometric Functions

The expression we need to find the sign for is \( \cos t \sec t \). First, recall the relationship between \( \sec t \) and \( \cos t \): \( \sec t = \frac{1}{\cos t} \). Therefore, \( \cos t \sec t = \cos t \times \frac{1}{\cos t} \).

Step 2: Simplify the Expression

Simplify the expression \( \cos t \times \frac{1}{\cos t} \). When multiplying these two expressions, the \( \cos t \) terms cancel each other out, resulting in \( 1 \).

Step 3: Determine the Sign

Since the expression simplifies to \( 1 \), which is a positive constant, it is always positive, regardless of the quadrant of \( t \). Therefore, the sign of \( \cos t \sec t \) is always positive.

Key Concepts

Cosine Function

The cosine function is a fundamental component of trigonometry. It represents the adjacent side's length over the hypotenuse's length in a right triangle. In terms of the unit circle, the cosine of an angle is the x-coordinate of a point on the circle.

• Cosine values range from -1 to 1.
• It is periodic, repeating every 360 degrees or \(2\pi\) radians.
• Cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.

Understanding when cosine is positive or negative helps determine the behavior of expressions involving cosine. In our exercise, knowing the quadrant helps predict cosine's sign; however, since we simplified our expression to a constant, this specific knowledge wasn't required to determine the sign of the final simplified expression.

Secant Function

The secant function is closely related to the cosine function. It is defined as the reciprocal of the cosine. Mathematically, it can be expressed as \( \sec t = \frac{1}{\cos t} \).

• Secant is undefined wherever cosine is zero because division by zero is undefined.
• Secant values can be greater than 1 or less than -1.
• It takes positive values in the first and fourth quadrants, similar to cosine.

When you are working with secant, it's important to remember its dependence on cosine. The sign of the secant matches that of the cosine when cosine is non-zero. This alignment plays a crucial role when determining trigonometric identities, as seen in our simplification exercise.

Trigonometric Simplification

Trigonometric simplification involves using identities to reduce complex expressions to simpler forms. These identities represent relationships between trigonometric functions, allowing for easier manipulation and evaluation.

Example of Simplification

Consider the expression \( \cos t \sec t \). Using the identity \( \sec t = \frac{1}{\cos t} \), we rewrite it as \( \cos t \times \frac{1}{\cos t} \).

• The cosine terms cancel each other out.
• The expression simplifies to 1, a straightforward constant.

Why Simplification Helps

Simplifying expressions helps to determine their properties, such as sign and value, easily without excessive calculations. In problems like the one described, finding a simplified constant like 1 tells us immediately that the expression's sign is always positive, facilitating quicker identification. Simplification is a powerful tool in trigonometry, making it easier to handle complex expressions efficiently.