Problem 198 - Precalculus

Question

For the following exercises, use reference angles to evaluate the expression. $$ \csc \frac{11 \pi}{6} $$

Step-by-Step Solution

Answer

The value of \( \csc \frac{11\pi}{6} \) is 2.

Step 1: Identify the Angle

The given angle is \( \frac{11\pi}{6} \). This angle is found in the fourth quadrant. It is more than \( 2\pi \), so it's effectively \( 2\pi - \frac{\pi}{6} \).

Step 2: Find the Reference Angle

A reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. For \( \frac{11\pi}{6} \), the reference angle is \( \frac{\pi}{6} \) since it is \( 2\pi - \frac{11\pi}{6} = \frac{\pi}{6} \).

Step 3: Calculate the Sine of the Reference Angle

The sine of the reference angle \( \frac{\pi}{6} \) is \( \frac{1}{2} \).

Step 4: Use the Reciprocal Identity

The cosecant function is the reciprocal of the sine function. Therefore, \( \csc \frac{11\pi}{6} = \frac{1}{\sin \frac{\pi}{6}} = \frac{1}{\frac{1}{2}} \).

Step 5: Simplify the Expression

Simplify the expression to get the value of the cosecant. \( \frac{1}{\frac{1}{2}} = 2 \).

Key Concepts

Reference Angles

Reference angles are crucial when working with trigonometric functions in different quadrants. A reference angle is always formed between the terminal side of an angle and the x-axis, essentially offering a mirror reflection toward a known axis for convenient calculation. This approach simplifies trigonometric calculations by reducing it to known values of angles in the first quadrant.

• Reference angles range from 0 to \( \frac{\pi}{2} \) radians or 0 to 90 degrees.
• They are always positive and less than or equal to \( \frac{\pi}{2} \).

For example, for the angle \( \frac{11\pi}{6} \), since it is in the fourth quadrant, the reference angle is \( \frac{2\pi}{6} - \frac{11\pi}{6} \) which equals \( \frac{\pi}{6} \). By identifying the reference angle, you can easily use known trigonometric values to determine function values.

Cosecant Function

The cosecant function, denoted as \( \csc \), is one of the main six trigonometric functions and serves as the reciprocal of the sine function. This means that for an angle \( \theta \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

• Cosecant is undefined whenever sine is zero because division by zero is not defined.
• The cosecant function is primarily used in the formulas of trigonometric identities and solving problems involving right triangles.

In the problem \( \csc \frac{11\pi}{6} \), by determining that \( \sin \frac{\pi}{6} = \frac{1}{2} \), we find that the cosecant of that reference angle is the reciprocal, \( \frac{1}{\frac{1}{2}} = 2 \). This use of reciprocal mathematics is central to trigonometry, offering powerful methods for problem-solving.

Reciprocal Identities

Reciprocal identities connect the primary trigonometric functions of sine, cosine, and tangent with their reciprocals, cosecant, secant, and cotangent, respectively. Understanding these identities is vital as they allow for transformations between angle measures.

• The fundamental reciprocal identities are: \( \csc(\theta) = \frac{1}{\sin(\theta)} \), \( \sec(\theta) = \frac{1}{\cos(\theta)} \), and \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
• These identities are often used alongside other trigonometric identities to simplify complex expressions and equations.

The exercise illustrates this by using the cosecant's reciprocal identity. To solve \( \csc \frac{11\pi}{6} \), we utilized the reciprocal identity for sine: \( \csc \theta = \frac{1}{\sin \theta} \). Through these transformations, solving trigonometric functions across different angles becomes more manageable and efficient.