Problem 186 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \csc \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The exact value of \( \csc \frac{\pi}{6} \) is 2.

Step 1: Convert Cosecant to Reciprocal of Sine

Recall that the cosecant function is the reciprocal of the sine function: \(\csc \theta = \frac{1}{\sin \theta}\). Thus, we need to find \(\sin \frac{\pi}{6}\) first, and then take its reciprocal.

Step 2: Evaluate Sine of \(\frac{\pi}{6}\)

The angle \(\frac{\pi}{6}\) is equivalent to \(30^{\circ}\). From trigonometric values, we know that \(\sin \frac{\pi}{6} = \sin 30^{\circ} = \frac{1}{2}\).

Step 3: Find Reciprocal of Sine Value

Since \(\csc \theta = \frac{1}{\sin \theta}\), we can substitute the sine value found into the expression for cosecant: \(\csc \frac{\pi}{6} = \frac{1}{\frac{1}{2}}\).

Step 4: Simplify the Expression

Simplify the reciprocal: \(\frac{1}{\frac{1}{2}} = 2\).

Key Concepts

Reciprocal Identities

Reciprocal identities are a key concept in trigonometry. They describe the relationship between certain pairs of trigonometric functions.
In essence, each of these pairs is one function expressed as the inverse, or reciprocal, of another.
Understanding reciprocal identities helps simplify complex trigonometric expressions and solve them more easily.
For instance:

• Cosecant (\(\csc \theta\)) is the reciprocal of sine (\(\sin \theta\)).
• Secant (\(\sec \theta\)) is the reciprocal of cosine (\(\cos \theta\)).
• Cotangent (\(\cot \theta\)) is the reciprocal of tangent (\(\tan \theta\)).

These relationships allow you to transform one trigonometric function into its counterpart, aiding in the computation of angles and side lengths in triangles. In our example, we can convert \(\csc \theta\) to the more familiar \(\sin \theta\), making it easier to find exact values.

Cosecant

Cosecant, denoted as\(\csc\), is one of the six primary trigonometric functions. It is defined as the reciprocal of the sine function.
This means that the cosecant of an angle\(\theta\) is calculated as\(\csc \theta = \frac{1}{\sin \theta}\).
Though not as commonly used as sine, cosine, and tangent, the cosecant function is crucial for various trigonometric calculations.
The key feature of the cosecant function is that it highlights values where the sine function approaches zero, thus highlighting asymptotic behavior in trigonometric analysis.
In practical terms, whenever you need \(\csc \theta\) you first evaluate \(\sin \theta\) and then take its reciprocal, as seen in the exercise where \(\csc \frac{\pi}{6} = 2\). This approach helps in accurately calculating values, especially in problems involving inverse trigonometric functions.

Sine

The sine function, often represented as\(\sin\), is one of the most fundamental concepts in trigonometry.
It describes the ratio of the opposite side to the hypotenuse in a right-angled triangle.
Mathematically, it's expressed as\(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
One of the primary attributes of the sine function is its periodicity and smooth waveform, which is integral in both theoretical studies and practical applications like signal processing.
Also, the sine function is closely related to the unit circle—a crucial geometric tool—which aids in determining sine values for any angle.
In our problem, for the angle \(\frac{\pi}{6}\), the sine value is \(\frac{1}{2}\). This demonstrates the utility of memorizing sine values for key angles used frequently, such as \(30^\circ\), \(45^\circ\), and \(60^\circ\). Knowing these values by heart helps with quick and efficient calculations.

Unit Circle

The unit circle is a central concept in trigonometry and is fundamental for understanding trigonometric functions and their values.
A unit circle is a circle with a radius of one, centered at the origin of a coordinate plane.
This circle provides a visual way to understand how the trigonometric functions relate to angles.

• The x-coordinate of a point on the unit circle at angle \(\theta\) represents \(\cos \theta\).
• The y-coordinate represents \(\sin \theta\).

Therefore, knowledge of the unit circle is essential for calculating sine, cosine, and other trigonometric functions.
When evaluating \(\sin \frac{\pi}{6}\) or any trigonometric values, the unit circle enables us to see that when the angle is \(\frac{\pi}{6}\) or \(30^\circ\), the y-coordinate is \(\frac{1}{2}\). This visual approach ties numeric values to geometric intuition, ensuring a deeper understanding of trigonometric principles.