Problem 87 - Precalculus: Functions and Graphs

Question

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\sin \theta=-\frac{5}{13} \text { and } \sec \theta>0$$

Step-by-Step Solution

Answer

The values are: \(\cos \theta = \frac{12}{13}\), \(\tan \theta = -\frac{5}{12}\), \(\sec \theta = \frac{13}{12}\), \(\csc \theta = -\frac{13}{5}\), \(\cot \theta = -\frac{12}{5}\).

Step 1: Analyze the Given Conditions

We are given that \( \sin \theta = -\frac{5}{13} \) and \( \sec \theta > 0 \). The condition that \( \sin \theta \) is negative indicates that \( \theta \) is in either the third or fourth quadrant. The condition \( \sec \theta > 0 \) means \( \cos \theta > 0 \), which places \( \theta \) in the fourth quadrant.

Step 2: Determine Cosine Using Pythagorean Identity

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find \( \cos \theta \). Substitute \( \sin \theta = -\frac{5}{13} \) into the identity:\[(-\frac{5}{13})^2 + \cos^2 \theta = 1\]\[\frac{25}{169} + \cos^2 \theta = 1\]\[\cos^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}\]Since \( \cos \theta > 0 \), we take the positive root:\( \cos \theta = \frac{12}{13} \).

Step 3: Calculate Other Trigonometric Functions

Now that we have \( \sin \theta = -\frac{5}{13} \) and \( \cos \theta = \frac{12}{13} \), we can calculate the other trigonometric functions:1. \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{5}{13}}{\frac{12}{13}} = -\frac{5}{12} \).2. \( \sec \theta = \frac{1}{\cos \theta} = \frac{13}{12} \).3. \( \csc \theta = \frac{1}{\sin \theta} = -\frac{13}{5} \).4. \( \cot \theta = \frac{1}{\tan \theta} = -\frac{12}{5} \).

Key Concepts

Quadrant Analysis

In trigonometry, understanding the quadrant in which an angle lies is crucial for determining the signs of its trigonometric functions. Each of the four quadrants of the Cartesian plane affects the sign.
In this exercise, we start by examining two conditions for the angle \( \theta \):

• \( \sin \theta = -\frac{5}{13} \) is negative, which tells us that \( \theta \) could either be in the third or fourth quadrant. This is because, in these quadrants, the sine function has a negative value.
• \( \sec \theta > 0 \) implies that \( \cos \theta > 0 \). Cosine is positive in the fourth quadrant, thus confirming that \( \theta \) lies here.

Quadrant analysis not only helps in identifying the sine of the function values but also guides the calculation of other trigonometric functions. It helps to only take positive values when required and decide the correct sign for solutions.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It relates the sine and cosine of an angle and is written as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is extremely useful for calculating unknown values when one of the trigonometric functions is known.

In the exercise, we're given \( \sin \theta = -\frac{5}{13} \). By substituting this into the Pythagorean Identity, we can solve for \( \cos \theta \).

• First, calculate \((-\frac{5}{13})^2 = \frac{25}{169}\).
• Next, substitute into the identity: \(\frac{25}{169} + \cos^2 \theta = 1\).
• Solve for \(\cos^2 \theta\): \(\cos^2 \theta = \frac{144}{169}\).
• Since \( \cos \theta > 0 \) in the fourth quadrant, take the positive root: \( \cos \theta = \frac{12}{13} \).

The Pythagorean Identity is a tool that makes solving trigonometric equations straightforward and manageable.

Trigonometric Functions Calculation

Once we determine \( \sin \theta \) and \( \cos \theta \), calculating the other trigonometric functions becomes straightforward using the primary trigonometric identities.

With \( \sin \theta = -\frac{5}{13} \) and \( \cos \theta = \frac{12}{13} \), the other functions can be calculated as follows:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{5}{13}}{\frac{12}{13}} = -\frac{5}{12} \)
• \( \sec \theta = \frac{1}{\cos \theta} = \frac{13}{12} \)
• \( \csc \theta = \frac{1}{\sin \theta} = -\frac{13}{5} \)
• \( \cot \theta = \frac{1}{\tan \theta} = -\frac{12}{5} \)

Each function derives from the relationships between them.
Understanding how trigonometric functions interconnect helps in solving complex problems and is fundamental for advanced applications. Trigonometric functions provide the tools to transition fluidly between them, translating initial known values into full function insights.