Problem 23 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \csc \frac{\pi}{2} $$

Step-by-Step Solution

Answer

The value of \(\csc(\frac{\pi}{2})\) is 1.

Step 1: Identify the trigonometric function to use

The exercise asks to find the value of \(\csc(\frac{\pi}{2})\), where \(\csc\) stands for cosecant, which is the reciprocal of the sine function. So, \(\csc(\theta) = \frac{1}{\sin(\theta)}\).

Step 2: Apply the trigonometric function

Here, we want to evaluate \(\csc(\frac{\pi}{2})\), but the cosecant is the reciprocal of the sine function. So we can rewrite the expression as \(\csc(\frac{\pi}{2}) = \frac{1}{\sin(\frac{\pi}{2})}\).

Step 3: Evaluate the sine function

According to the unit circle, we know that that \(sin(\frac{\pi}{2}) = 1\). So we can substitute this value into the previous equation and get \(\csc(\frac{\pi}{2}) = \frac{1}{1}\).

Step 4: Simplify the expression

Simplify the expression \(\frac{1}{1}\), which results in 1.

Key Concepts

Cosecant

To fully understand the concept of cosecant, firstly recognize that it’s one of the six fundamental trigonometric functions. The cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the opposite side. However, a lot of times, we think about trigonometric functions in terms of the unit circle.

• The cosecant function is represented as \( \csc \theta \).
• It is the reciprocal of the sine function, meaning it is the inverse relationship.
• Cosecant is undefined for angles where the sine is zero since dividing by zero is undefined.

For an angle \( \theta \) measured in radians, the cosecant function is calculated by \( \csc \theta = \frac{1}{\sin \theta} \). It is important to remember this reciprocal relationship when solving trigonometric expressions that involve cosecant.

Sine Function

The sine function is a cornerstone in trigonometry. It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
Sine is often depicted using the unit circle, where parameters of trigonometric functions are governed by angle measures.

• Sine represents a smooth, periodic oscillation.
• In the unit circle, the value of sine corresponds to the y-coordinate.
• The sine function ranges between -1 and 1.

In mathematical terms, \( \sin \theta \) can be computed directly from the angle \( \theta \). For example, \( \sin(\frac{\pi}{2}) = 1 \) because at \( \frac{\pi}{2} \), the angle makes the maximum height in the unit circle. Understanding sine’s properties allows for efficient computation of related functions like cosecant.

Reciprocal Identities

Reciprocal identities provide great analytical power when working with trigonometric expressions. They express relationships between different trigonometric functions through inverses.

• Reciprocal identities include relationships like \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).
• These identities help transform complex trigonometric equations into simpler, solvable forms.
• Using reciprocal identities simplifies finding values for functions that might not be directly calculable at first glance.

By learning and applying reciprocal identities, such as transforming \( \csc(\theta) \) into \( \frac{1}{\sin(\theta)} \), you leverage known values of sine to determine cosecant, as demonstrated in evaluating \( \csc(\frac{\pi}{2}) \) from \( \sin(\frac{\pi}{2}) \). Mastery over these identities empowers you to tackle a wide array of trigonometric challenges effectively.