Problem 20 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \csc 45^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \csc(45^{\circ}) \) is \( \sqrt{2} \).

Step 1: Understand the definition of cosecant function

The cosecant of an angle is defined as the reciprocal of its sine function. This means, \[ \csc(\theta) = \frac{1}{\sin(\theta)} \]. Therefore, to find the value of \( \csc(45^{\circ}) \), we need to find the reciprocal of the sine of 45 degrees.

Step 2: Calculate sine of 45 degrees

From the values of sine function that are known, we have \( \sin(45^{\circ}) = \sqrt{2}/2 \)

Step 3: Calculate cosecant of 45 degrees

As cosecant of theta is equal to the reciprocal of sine of theta, we have \( \csc(45^{\circ}) = \frac{1}{\sin(45^{\circ})} \). Substituting \( \sin(45^{\circ}) = \sqrt{2}/2 \), we get \( \csc(45^{\circ}) = \frac{1}{\sqrt{2}/2} \), which simplifies to \( \csc(45^{\circ}) = \sqrt{2} \).

Key Concepts

Cosecant Function

Understanding trigonometric functions, especially the cosecant function, is crucial in trigonometry. The cosecant function, denoted by \( \csc(\theta) \), is a bit unique because it's the reciprocal of the sine function. Simply put, for an angle \( \theta \), the cosecant function is calculated as:

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

This means that whenever you're asked to find the cosecant of an angle, your first step is to determine the sine of that angle.
Then, take the reciprocal of that value to get the cosecant. It's important to remember that the sine function cannot be zero here, as the reciprocal of zero would be undefined. Therefore, understanding both the sine and cosecant functions is critical when solving trigonometric problems.
Now, remembering the mathematical definition alone might not be enough. What makes it easier is practice and knowing the common angles for which values are typically memorized, like \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), and \( 90^\circ \). In our example, you can quickly find \( \csc(45^\circ) \) once you know \( \sin(45^\circ) \). With more practice using these reciprocal relationships, you will get more comfortable with solving such problems.

Sine Function

The sine function is one of the primary trigonometric functions, often introduced first in the study of right triangles. It's an essential part of trigonometry, helping us understand the ratios of different sides of a right triangle. Specifically, for any angle \( \theta \) in a triangle, the sine of \( \theta \) is defined as:

• \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)

Moreover, there are particular angles like \( 0^\circ \), \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), and \( 90^\circ \) whose sine values are well-established and memorized.
For example, \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \). This value is derived from considering an isosceles right triangle with an angle of \( 45^\circ \). In such triangles, both opposite and adjacent sides are equal, leading to this common sine value.
Knowing these sine values is instrumental when calculating the cosecant, as it is directly the reciprocal. If you remember, for the sine function:

• \( \sin(0^\circ) = 0 \)
• \( \sin(30^\circ) = \frac{1}{2} \)
• \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)
• \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
• \( \sin(90^\circ) = 1 \)

These become invaluable as reference points for solving trigonometric functions involving the sine in various problems.

Angle Measurement

Understanding how to measure angles is foundational in trigonometry. Angles can be measured in degrees or radians, with each having its own usage context. In the exercise you're dealing with, angles are given in degrees, which is a common measurement in many fields and everyday applications.
A full circle is comprised of \( 360^\circ \) degrees. This means half a circle is \( 180^\circ \), and a quarter of a circle is \( 90^\circ \). These fractions of a circle are often used when working with trigonometric functions, as certain angles have well-known values, such as the 45 degrees used in the exercise.
When working with angles in degrees:

• Know common angles and their sine values.
• Understand acute angles (between \( 0^\circ \) and \( 90^\circ \)).
• Works within quadrants: 0 to 90, 90 to 180, etc.

These angles lend themselves to predicting behavior of sine and cosecant functions effectively. In mathematical contexts beyond simple exercises, knowing how to convert between degrees and radians is also significant as radians are frequently used in advanced mathematics and physics. However, in problems like these, focusing on degrees keeps the calculations straightforward and connects directly to the sine values you have memorized. This awareness aids not only in exams but also in practical fields such as engineering and navigation, where precise angle measurement correlates to specific outcomes.