Problem 56 - Precalculus

Question

For the following exercises, simplify the given expression. $$ \csc \left(\frac{\pi}{2}-t\right) $$

Step-by-Step Solution

Answer

\( \sec(t) \)

Step 1: Recognize Trigonometric Identities

The expression involves a trigonometric function, specifically the cosecant function, and an angle subtraction: \( \csc \left( \frac{\pi}{2} - t \right) \). To simplify this, we need to use trigonometric identities.

Step 2: Use the Co-function Identity

For trigonometric functions, co-function identities relate functions with angles that add to \( \frac{\pi}{2} \). The co-function identity for sine and cosecant is \( \csc \left( \frac{\pi}{2} - t \right) = \sec(t) \).

Step 3: Write the Simplified Expression

Using the identity from the previous step, we find that \( \csc \left( \frac{\pi}{2} - t \right) \) simplifies to \( \sec(t) \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(\theta) \), is one of the six fundamental trigonometric functions, and it's the reciprocal of the sine function. In simple terms, it is defined as:

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

This means that wherever the sine of an angle is zero, the cosecant function will be undefined, as division by zero is not possible.

Understanding the behavior of the cosecant function is crucial for solving trigonometric problems and simplifying expressions.
If you think of the unit circle, where each point represents an angle, the cosecant gives you the vertical stretch based on how far the sine value takes you, but in reverse, on the unit circle. Every angle has a corresponding cosecant value, as long as the sine is not zero.

Co-function Identities

Co-function identities are trigonometric identities that relate two functions with angles that add up to \( \frac{\pi}{2} \) (or \( 90^{\circ} \)). They are important because they help in transforming expressions and simplifying complex trigonometric functions.For example, the co-function identity involving sine and cosecant, and how it was applied to this exercise is:

• \( \csc \left( \frac{\pi}{2} - t \right) = \sec(t) \)
• Similarly, \( \sin \left( \frac{\pi}{2} - t \right) = \cos(t) \)

This identity helps in expressing one trigonometric function into another, which can be extremely useful during simplification processes. It's about relating angles and finding counterpart functions in a different form. Co-function identities are crucial to know because they allow flexibility and creativity in problem-solving, making different problems simpler to manage.

Angle Subtraction

Angle subtraction involves manipulating the angles within trigonometric functions, often to apply various trigonometric identities for simplification.

For instance, in our original expression \( \csc \left( \frac{\pi}{2} - t \right) \), the angle \( \left( \frac{\pi}{2} - t \right) \) is a form of angle subtraction. The subtraction here indicates the usage of co-function identities because it represents an angle transformation within the domain of trigonometric functions.

• Angle subtraction can simplify an expression by allowing you to transition between what might seem different, but is mathematically equivalent.
• This technique is particularly helpful in changing angles so that they align with the standard angle positions in trigonometry, e.g., \( 0, \frac{\pi}{2}, \pi, \) etc.

Learning to recognize when angle subtraction is present and useful can simplify many mathematical operations involving trigonometric functions, making broader applications and problem solving more intuitive.