Problem 12 - Calculus Single Variable

Question

Calculate the given expression without using a calculator. \(\sec (-\pi / 3)^{\csc (-\pi / 2)}\)

Step-by-Step Solution

Answer

The value of the expression is \( \frac{1}{2} \).

Step 1: Understanding the Problem

We need to evaluate the expression \( \sec (-\pi / 3)^{\csc (-\pi / 2)} \). This involves finding the values of \( \sec(-\pi/3) \) and \( \csc(-\pi/2) \) separately.

Step 2: Evaluate \( \sec(-\pi/3) \)

The secant function is the reciprocal of the cosine function. Therefore, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Since cosine is an even function, \( \cos(-\theta) = \cos(\theta) \), we have \( \sec(-\pi/3) = \sec(\pi/3) \). \( \cos(\pi/3) = \frac{1}{2} \), so \( \sec(\pi/3) = \frac{1}{\frac{1}{2}} = 2 \). Hence, \( \sec(-\pi/3) = 2 \).

Step 3: Evaluate \( \csc(-\pi/2) \)

The cosecant function is the reciprocal of the sine function, thus \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Since sine is an odd function, \( \sin(-\theta) = -\sin(\theta) \), we have \( \sin(-\pi/2) = -\sin(\pi/2) = -1 \). Therefore, \( \csc(-\pi/2) = \frac{1}{-1} = -1 \).

Step 4: Combine the Values

Now, we substitute the values found into the original expression: \( \sec(-\pi/3)^{\csc(-\pi/2)} = 2^{-1} \).

Step 5: Simplify the Expression

The expression \( 2^{-1} \) means the reciprocal of 2, which is \( \frac{1}{2} \). Therefore, the final value of the expression is \( \frac{1}{2} \).

Key Concepts

Secant Function

The secant function is related closely to the cosine function in trigonometry. It is one of the six fundamental trigonometric functions, each playing crucial roles in different mathematical contexts.

Unlike the cosine function, the secant is its reciprocal. That means if you're looking at the secant of an angle \( \theta \), it's the inverse of what cosines provide. It's defined as:\[\sec(\theta) = \frac{1}{\cos(\theta)}\]
This relationship is foundational. The secant function can often give you quick insights into problems dealing with angles and lengths where direct calculations via the cosine aren’t feasible.

Let's break it down:

• It doesn’t exist wherever cosine is zero (because you can’t divide by zero).
• It's a periodic function like all trigonometric functions, repeating its values in a regular pattern.
• This function becomes particularly useful in complex calculations, helping us to find solutions efficiently.

By its nature, understanding secant often serves as a stepping stone to more complex trigonometric functions and deeper mathematical truths.

Cosecant Function

The cosecant function complements the sine function and offers a similar reciprocal relationship as the secant has with cosine. When you hear the term `cosecant`, think of the sine flipped over. It is defined as:\[\csc(\theta) = \frac{1}{\sin(\theta)}\]
This relationship is crucial when working in trigonometry as the sine function can vary from -1 to 1. The cosecant, therefore, extends from negative to positive infinity excluding values where the sine equals zero.

Let's find out why it's important:

• It's undefined at every point where the sine is zero, avoiding division by zero errors.
• Cosecant helps solve trigonometric problems classically by making reciprocal relationships clear.
• It shares a periodic nature with the sine function, which becomes evident every \(2\pi\) radians.

Applying the cosecant strategically allows for elegant solutions to complex angle problems, such as the one evaluated here. Keep in mind the periodic and reciprocal properties when using it.

Even and Odd Functions

Two intriguing properties of trigonometric functions are being `even` or `odd`. These classifications tell us how the function behaves when you switch the sign of its input, \(\theta\).

The cosine and secant functions are both even functions. This means their value doesn't change whenever the input sign is switched:\[\cos(-\theta) = \cos(\theta) \quad \text{and} \quad \sec(-\theta) = \sec(\theta)\]
In simpler terms, they exhibit perfect symmetry along the y-axis of their graph. This property makes solving trigonometric problems a bit easier, as seen with \(\sec(-\pi/3)\) simplifying directly to \(\sec(\pi/3)\).

Conversely, sine and cosecant are odd functions, where the flip in sign of the input results in a sign flip in the output:\[\sin(-\theta) = -\sin(\theta) \quad \text{and} \quad \csc(-\theta) = -\csc(\theta)\]
This trait means they are symmetric relative to the origin on a graph.

Knowing whether a function is even or odd helps quickly assess the nature of trigonometric expressions and solve problems by understanding their inherent symmetrical properties.