Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact value. (a) \(\csc (3 \pi / 4)\) (b) \(\csc (-2 \pi / 3)\)

Step-by-Step Solution

Answer

(a) \(\sqrt{2}\); (b) \(-\frac{2\sqrt{3}}{3}\)

Step 1: Understanding the cosecant function

The cosecant function is the reciprocal of the sine function. Therefore, \( \csc(\theta) = \frac{1}{\sin(\theta)} \). To find \( \csc(\theta) \), we need to determine the value of \( \sin(\theta) \).

Step 2: Determine \( \sin(3\pi/4) \)

The angle \( 3\pi/4 \) is in the second quadrant. In the unit circle, \( \sin(\theta) = \sin(\pi - \theta) \). So \( \sin(3\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2} \).

Step 3: Calculate \( \csc(3\pi/4) \)

Given \( \sin(3\pi/4) = \frac{\sqrt{2}}{2} \), the cosecant is \( \csc(3\pi/4) = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \).

Step 4: Determine \( \sin(-2\pi/3) \)

The angle \( -2\pi/3 \) is in the fourth quadrant, where sine is negative. \( \sin(-2\pi/3) = -\sin(2\pi/3) \). In the unit circle, \( \sin(2\pi/3) = \sin(\pi/3) = \frac{\sqrt{3}}{2} \), so \( \sin(-2\pi/3) = -\frac{\sqrt{3}}{2} \).

Step 5: Calculate \( \csc(-2\pi/3) \)

Given \( \sin(-2\pi/3) = -\frac{\sqrt{3}}{2} \), the cosecant is \( \csc(-2\pi/3) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \) after rationalizing the denominator.

Key Concepts

Cosecant Function

The cosecant function is an important trigonometric function and is represented as \( \csc(\theta) \). It's simply the reciprocal of the sine function. This means:

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

Understanding this relationship means that whenever you need to find the cosecant of an angle, you start by finding its sine value and then take the reciprocal.
The cosecant is undefined whenever the sine of an angle is zero, as dividing by zero is not possible. Always check quadrant positions and sine values to ensure accurate calculations.

Unit Circle

The unit circle is a handy tool in trigonometry and is used widely to determine the values of trigonometric functions at various angles. It’s a circle with a radius of one, which helps simplify calculations.

• Each point on the unit circle corresponds to coordinates \((\cos(\theta), \sin(\theta))\).
• Angles are measured from the positive x-axis, moving counterclockwise.

Key positions on the unit circle include angles like \( \pi/4, \pi/3, \pi/2 \), and so on. When working with the unit circle, always pay attention to the quadrant in which the angle lies:
- First quadrant: All sine values are positive.
- Second quadrant: Sine values remain positive but cosine values turn negative.
Understanding which function is positive or negative depending on the quadrant is crucial for obtaining the correct trigonometric values.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include cosecant, secant, and cotangent. Each has its corresponding primary function:

• Cosecant \( \csc(\theta) = \frac{1}{\sin(\theta)}\)
• Secant \( \sec(\theta) = \frac{1}{\cos(\theta)}\)
• Cotangent \( \cot(\theta) = \frac{1}{\tan(\theta)}\)

Learning these functions is crucial because they frequently appear in problems and real-world applications. They help in simplifying equations and solving for angles and distances that aren’t directly measurable.
Always be careful with signs and quadrant positions, these affect whether the final value is positive or negative. Rationalizing denominators is often necessary for clean, simplified results.

Sine Function

The sine function is fundamental in trigonometry and appears in various mathematical and real-world contexts. It's defined as the y-coordinate of a point on the unit circle associated with an angle \( \theta \), given as \( \sin(\theta) \).

• The sine of an angle measures the vertical component of its corresponding point.
• It helps in determining the height in right-angle triangles.

The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants. Recognizing these sign changes is necessary for precise calculations when using angles in different quadrants.
Common angles you often use are \( \pi/6, \pi/4, \pi/3 \), and knowing these values makes finding reciprocals like the cosecant a breeze.