Problem 2 - Algebra 2

Question

Find the value of each expression. $$ \csc \theta, \text { if } \cos \theta=-\frac{3}{5} ; 180^{\circ} \leq \theta<270^{\circ} $$

Step-by-Step Solution

Answer

\( \csc \theta = -\frac{5}{4} \)

Step 1: Identify the quadrant

Since the angle \( \theta \) is in the range \( 180^{\circ} \leq \theta < 270^{\circ} \), it means that \( \theta \) is in the third quadrant.

Step 2: Determine the sign of the sine function

In the third quadrant, both sine and cosine are negative. Therefore, \( \sin \theta \) will be negative.

Step 3: Use the Pythagorean identity

The identity \( \sin^2 \theta + \cos^2 \theta = 1 \) allows us to find the sine of the angle. Substituting the given \( \cos \theta = -\frac{3}{5} \), we have: \( \sin^2 \theta + \left( -\frac{3}{5} \right)^2 = 1 \).

Step 4: Solve for \( \sin \theta \)

Let's calculate it:1. \( \sin^2 \theta + \frac{9}{25} = 1 \)2. \( \sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \)3. \( \sin \theta = \pm \frac{4}{5} \)Since \( \sin \theta \) is negative in the third quadrant, \( \sin \theta = -\frac{4}{5} \).

Step 5: Find \( \csc \theta \)

The cosecant function is the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \). Substituting for \( \sin \theta \), we get \( \csc \theta = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine functions. It's derived from the Pythagorean Theorem for a unit circle, where the radius is 1. The identity is expressed as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This equation holds for all values of \( \theta \) and helps us calculate trigonometric functions when one of them is known. In the original problem, it allowed us to find \( \sin \theta \) given \( \cos \theta \) was known. By substituting \( \cos \theta = -\frac{3}{5} \), you can determine that:- \( \sin^2 \theta = \frac{16}{25} \)- Since \( \theta \) is in the third quadrant where sine is negative, \( \sin \theta = -\frac{4}{5} \).The Pythagorean Identity is a critical tool for solving trigonometric equations and proving identities, making it an essential part of any mathematician's toolkit.

Sine Function

The sine function, commonly denoted as \( \sin \theta \), refers to the ratio of the opposite side to the hypotenuse in a right triangle. It is one of the primary trigonometric functions. Importantly, the sine function changes signs depending on the quadrant of the angle \( \theta \).Here's a quick guide:

• 1st Quadrant: Sine is positive.
• 2nd Quadrant: Sine is positive.
• 3rd Quadrant: Sine is negative.
• 4th Quadrant: Sine is negative.

In the context of the original exercise, \( \theta \) is situated in the third quadrant (\(180^{\circ} \leq \theta < 270^{\circ}\)), meaning the sine value is negative. This piece of information helped to solve for \( \sin \theta = -\frac{4}{5} \) using the given \( \cos \theta \) and the Pythagorean Identity. The understanding of which quadrants affect the sign of the function is crucial when calculating trigonometric values.

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. It is calculated as:\[\csc \theta = \frac{1}{\sin \theta} \]Cosecant is considered a secondary trigonometric function but is still quite useful, particularly when dealing with values of sine that are not zero—since division by zero is undefined.In the problem provided, once \( \sin \theta \) was determined to be \(-\frac{4}{5}\), finding \( \csc \theta \) became straightforward:- \( \csc \theta = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \).The understanding of reciprocal relationships between sine and cosecant functions assists in a broader comprehension of function behavior in trigonometry, broadening the scope of solving more complex trigonometric problems efficiently.