Problem 13 - Precalculus

Question

Find the exact values of the indicated trigonometric functions using the unit circle. $$\csc \left(\frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

\(\csc\left(\frac{5\pi}{6}\right) = 2\).

Step 1: Identify the Reference Angle

The given angle is \(\frac{5\pi}{6}\). First, recognize that it falls in the second quadrant of the unit circle. The reference angle is calculated as \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).

Step 2: Identify the Sine Value Using Reference Angle

For the reference angle \(\frac{\pi}{6}\), its sine value comes from the unit circle as \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\). In the second quadrant, the sine function is positive, so \(\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}\).

Step 3: Calculate the Cosecant Value

The cosecant function is the reciprocal of the sine function. Therefore, \(\csc\left(\frac{5\pi}{6}\right) = \frac{1}{\sin\left(\frac{5\pi}{6}\right)}\). Substitute the sine value: \(\csc\left(\frac{5\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of the coordinate plane. The unit circle allows us to easily determine the values of trigonometric functions for various angles.

• Key Points:
  • The x-coordinate of a point on the unit circle represents the cosine of the angle.
  • The y-coordinate represents the sine of the angle.
  • The angle is measured from the positive x-axis, counterclockwise.

Using the unit circle, you can find exact values for sine, cosine, and other trigonometric functions without a calculator. This is particularly useful in mathematics, physics, and engineering contexts.

Cosecant

Cosecant is one of the less commonly used trigonometric functions, but it is essential for a comprehensive understanding of trigonometry. It is defined as the reciprocal of the sine function.

• Formula:
  • \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Characteristics:
  • Cosecant is undefined for angles where sine is zero (like 0, \(\pi\), 2\(\pi\)).
  • It always has the same sign as the sine function.

Understanding the cosecant function complements your knowledge of the unit circle, as it directly relates to the sine ratios observed in the circle's layout.

Reference Angle

A reference angle is crucial for solving trigonometric problems, especially when dealing with angles greater than \(90^{\circ}\) or \(\pi/2\). It helps in transferring known angle values to other quadrants.

• Definition: The reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis.
• Calculation:
  • For angles in the second quadrant, \(\text{Reference Angle} = \pi - \text{angle} \).
  • For third and fourth quadrants, different formulas apply for calculation based on the quadrant.

By knowing your reference angle, you can determine the original angle's sine, cosine, and tangent based on this simpler, smaller angle.

Reciprocal Functions

Reciprocal functions in trigonometry include cosecant, secant, and cotangent, derived from the primary trigonometric functions. They help to provide a deeper understanding of the inherent relationships in trigonometry.

• Examples:
  • Cosecant: \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
  • Secant: \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
  • Cotangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• Application:
  • Useful for solving complex trigonometric equations.
  • Help in understanding angles and their ratios on the unit circle.

Reciprocals are great for fast calculations and broadening your mathematical toolkit when dealing with trigonometric functions.