Problem 90 - Precalculus: Functions and Graphs

Question

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\csc \theta=5 \text { and } \cot \theta<0$$

Step-by-Step Solution

Answer

Find \( \sin \theta = \frac{1}{5} \), \( \cos \theta = -\frac{\sqrt{24}}{5} \), and other trigonometrics.

Step 1: Identify Given Information

We are given that \( \csc \theta = 5 \), which means \( \theta = \frac{1}{\sin \theta} \). Additionally, \( \cot \theta < 0 \) indicates that \( \theta \) is in the second or fourth quadrant since the cotangent is negative in those quadrants.

Step 2: Find Sin θ Using Cosecant Definition

Since \( \csc \theta = 5 \), it follows from the definition \( \csc \theta = \frac{1}{\sin \theta} \) that \( \sin \theta = \frac{1}{5} \).

Step 3: Determine the Appropriate Quadrant

Using \( \cot \theta < 0 \), we know that since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), the cosine must be negative, implying \( \theta \) is in the second quadrant where \( \sin \) is positive and \( \cos \) is negative.

Step 4: Use Pythagorean Identity to Find Cos θ

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substituting \( \sin \theta = \frac{1}{5} \), we get \( \left(\frac{1}{5}\right)^2 + \cos^2 \theta = 1 \). Simplifying gives \( \cos^2 \theta = 1 - \frac{1}{25} = \frac{24}{25} \), so \( \cos \theta = -\sqrt{\frac{24}{25}} = -\frac{\sqrt{24}}{5} \) because \( \cos \theta \) is negative in quadrant II.

Step 5: Calculate Remaining Trigonometric Functions Using Found Values

Now calculate the other trigonometric functions:- \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{5}}{-\frac{\sqrt{24}}{5}} = -\frac{1}{\sqrt{24}} \)- \( \sec \theta = \frac{1}{\cos \theta} = -\frac{5}{\sqrt{24}} \) - Given \( \csc \theta = 5 \).- \( \cot \theta = \frac{1}{\tan \theta} = \sqrt{24} \)

Key Concepts

Cosecant

Cosecant, abbreviated as \( \csc \), is a trigonometric function that is the reciprocal of the sine function. If \( \sin \theta \) represents the sine of angle \( \theta \), then \( \csc \theta = \frac{1}{\sin \theta} \). This means that to find the cosecant of an angle, you simply take the inverse of its sine value.

This concept is particularly useful in situations where you already have the value of \( \csc \theta \) and need to find \( \sin \theta \). For instance, if you know \( \csc \theta = 5 \), you can easily find that \( \sin \theta = \frac{1}{5} \).

• Allows conversion between sine and cosecant functions using a reciprocal relationship.
• Helps in solving trigonometric equations where \( \sin \theta \) is needed but only \( \csc \theta \) is known.

Cosecant is particularly significant in trigonometry for understanding the relationships between different trigonometric functions and in triangle calculus, where the sides of a right triangle are involved.

Quadrant Analysis

Quadrant analysis is essential in trigonometry to determine the sign and behavior of trigonometric functions depending on the angle's position. The coordinate plane is divided into four quadrants, and each quadrant has implications for the sign of sine, cosine, tangent, and their reciprocals.

To determine which quadrant an angle falls into, it's crucial to use any given conditions about the trigonometric values. In our scenario, \( \cot \theta < 0 \), which means that the angle \( \theta \) is in a quadrant where the cosine and sine have opposite signs.

• In the second quadrant, \( \sin \theta \) is positive and \( \cos \theta \) is negative.
• In the fourth quadrant, \( \sin \theta \) is negative and \( \cos \theta \) is positive.

Since we know \( \csc \theta = 5 \), implying \( \sin \theta > 0 \), and \( \cot \theta \) is negative, \( \theta \) must be in the second quadrant, where these conditions meet.

Quadrant analysis not only allows us to find out where an angle lies but also facilitates the prediction of the positivity or negativity of other trigonometric functions.

Pythagorean Identity

The Pythagorean identity is one of the fundamental relationships in trigonometry and is used to connect the sine and cosine functions. The most well-known form is \( \sin^2 \theta + \cos^2 \theta = 1 \).

This identity is particularly useful when one trigonometric function is known, and you need to determine another. For example, when you know \( \sin \theta \) (from the cosecant \( \csc \theta = 5 \), we have \( \sin \theta = \frac{1}{5} \)), you can find \( \cos \theta \) by substituting into the identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( \left(\frac{1}{5}\right)^2 + \cos^2 \theta = 1 \)
• \( \cos^2 \theta = 1 - \frac{1}{25} = \frac{24}{25} \)

From here, take the square root to find \( \cos \theta \), ensuring to apply the correct sign based on quadrant analysis (\( \cos \theta = -\frac{\sqrt{24}}{5} \) in the second quadrant).

This identity allows us to solve for unknowns across numerous trigonometric problems, providing a strong backbone for understanding angles and their corresponding function values.