Problem 44 - Algebra 2

Question

Find the exact value of each trigonometric function. $$ \csc \left(\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

\( \csc \left(\frac{\pi}{4}\right) = \sqrt{2} \)

Step 1: Understanding the Cosecant Function

The cosecant function, \( \csc \theta \), is defined as the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \). To find \( \csc \left(\frac{\pi}{4}\right) \), we first need to determine \( \sin \left(\frac{\pi}{4}\right) \).

Step 2: Calculate \( \sin \left(\frac{\pi}{4}\right) \)

\( \frac{\pi}{4} \) radians corresponds to \( 45^\circ \). The sine of \( 45^\circ \) is \( \frac{\sqrt{2}}{2} \). Thus, \( \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 3: Find the Reciprocal to get \( \csc \left(\frac{\pi}{4}\right) \)

Now that we know \( \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we calculate the cosecant by taking the reciprocal: \( \csc \left(\frac{\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} \).

Step 4: Simplify the Expression

To simplify \( \frac{1}{\frac{\sqrt{2}}{2}} \), multiply the numerator and the denominator by \( 2 \) to rationalize the denominator. This gives \( \csc \left(\frac{\pi}{4}\right) = \frac{2}{\sqrt{2}} = \sqrt{2} \).

Key Concepts

Cosecant

Cosecant, often abbreviated as \( \csc \), is one of the six fundamental trigonometric functions. Primarily, it is known as the reciprocal of the sine function. This means that for any angle \( \theta \), the cosecant is defined as:

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

Cosecant is particularly useful in trigonometry when dealing with right-angled triangles and the unit circle. It helps in finding values related to angles and sides where the sine is very small or zero is not permissible.
When working with angles expressed in radians, the utility of the cosecant function becomes indispensable as it complements the sine function by providing an alternative perspective. To ensure accuracy, it’s crucial to always begin by correctly determining the sine of the given angle.

Sine Function

The sine function, denoted as \( \sin \), is a vital concept in trigonometry. It represents the ratio of the length of the opposite side to the hypotenuse in a right triangle:

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

This function is crucial for calculating various properties in mathematics and engineering fields. It becomes even more critical when dealing with the concept of the unit circle, where the radius (hypotenuse) is always one unit, simplifying calculations considerably.
For instance, to find \( \sin \left(\frac{\pi}{4}\right) \), one can relate \( \frac{\pi}{4} \) radians to \( 45^\circ \) degrees. The sine of \( 45^\circ \) is well-known and equals \( \frac{\sqrt{2}}{2} \). This knowledge forms the base for further trigonometric calculations, like determining its reciprocal in the cosecant function.

Reciprocal of Sine

The concept of the reciprocal of sine introduces the idea of inverses in trigonometry. Understanding reciprocals, like in the case of the cosecant function, is about flipping the ratio:

• Given \( \sin \theta = \frac{a}{b} \), its reciprocal \( \csc \theta = \frac{b}{a} \)

The reciprocal function is essential in contexts where the original sine value is small, and divisions become non-intuitive. For \( \frac{\pi}{4} \) radians, or \( 45^\circ \), \( \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Consequently, its reciprocal \( \csc \left(\frac{\pi}{4}\right) \) is derived by flipping the sine result to obtain \( \sqrt{2} \).
This inversion assists in distinguishing various aspects of angles, facilitating deeper understanding and problem-solving techniques in trigonometry.

Radians

Radians measure angles based on the radius of a circle. It is an alternative to degrees, commonly used in calculus and advanced mathematics. One full circle in radians equates to \( 2\pi \), while in degrees it’s \( 360^\circ \):

• \( 1 \text{ radian} \approx 57.3^\circ \)
• \( \pi \approx 3.1416 \)

Radians provide a natural and compact measure of angles, allowing trigonometric formulas and identities to present more gracefully and succinctly.
For instance, \( \frac{\pi}{4} \) radians are equivalent to \( 45^\circ \). Radians are most often used in settings involving rotations, periodic functions, and wave equations, providing coherent and uniform measurements for angular velocity and frequency.
Embracing radians smooths the transition to advanced topics while ensuring that foundational trigonometric understanding remains robust.