Problem 34 - Precalculus

Question

Find the indicated trigonometric function values. If \(\sec \theta=-\frac{13}{5}\) and the terminal side of \(\theta\) lies in quadrant II, find csc \(\theta\)

Step-by-Step Solution

Answer

\( \csc \theta = \frac{13}{12} \)

Step 1: Understand Given Values and Quadrant

We are given that \( \sec \theta = -\frac{13}{5} \) and \( \theta \) is in the second quadrant. In the second quadrant, the cosine function is negative, which matches the negative secant value since \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Calculate \( \cos \theta \)

Using the relation \( \sec \theta = \frac{1}{\cos \theta} \), we substitute the given secant value to find \( \cos \theta \): \( \cos \theta = \frac{1}{\sec \theta} = \frac{1}{-\frac{13}{5}} = -\frac{5}{13} \).

Step 3: Use Pythagorean Identity to Find \( \sin \theta \)

The Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) helps us find \( \sin \theta \). Substitute \( \cos \theta = -\frac{5}{13} \):\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} \].Thus, \( \sin \theta = \pm \frac{12}{13} \). In the second quadrant, sine is positive, so \( \sin \theta = \frac{12}{13} \).

Step 4: Calculate \( \csc \theta \)

Now, find \( \csc \theta \) using the sine value: \( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12} \).

Key Concepts

Secant Function

The secant function, abbreviated as \( \sec \), is a fundamental trigonometric function, commonly used in various applications. It is essentially the reciprocal of the cosine function.

• Defined as \( \sec \theta = \frac{1}{\cos \theta} \), secant gives insight into the relationship between an angle \( \theta \) and the coordinates of its terminal side in a unit circle.
• This function is particularly useful in trigonometry when solving problems involving right triangles or the unit circle, especially when you need information that involves division by the horizontal axis.
• One important thing to note about the secant function is its behavior in different quadrants: in quadrants where cosine is positive (I and IV), secant is also positive, whereas it is negative in quadrants where cosine is negative (II and III).

Understanding the secant function helps solve exercises that involve other trigonometric relationships, especially in determining unknown side lengths and angles, as seen in the given exercise, where knowing the secant value helped find the cosecant.

Cosecant Function

The cosecant function, abbreviated as \( \csc \), is another vital trigonometric function that's often encountered in mathematics.

• Cosecant is the reciprocal of the sine function, defined as \( \csc \theta = \frac{1}{\sin \theta} \).
• This function is particularly helpful when you've already determined the sine of an angle and need to find a related trigonometric ratio, such as in computations involving wave equations or harmonic motion.
• Much like secant, the sign of the cosecant function is determined by the quadrant in which the angle lies: cosecant is positive where sine is positive, which occurs in quadrants I and II, and negative in quadrants where sine is negative (III and IV).

In our context, knowing the quadrant allowed us to determine the sign of the sine function, which in turn helped compute the value of \( \csc \theta \). This was essential to solving the exercise correctly.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometric relations, elegantly binding sine and cosine functions in a simple yet powerful equation.

• It is given by \( \sin^2 \theta + \cos^2 \theta = 1 \).
• This identity stems from the Pythagorean theorem applied in the context of the unit circle, where each radial line segment can be described using sine and cosine.
• Leveraging the Pythagorean identity allows for the simplification and solving of trigonometric equations, as you can express either sine or cosine in terms of the other to find missing values.

In the provided exercise, using the Pythagorean identity was key: it enabled us to find \( \sin \theta \) from the known \( \cos \theta \). The identity helps to ensure that all trigonometric functions are consistent with each other, maintaining the harmony of the unit circle relationships. This insight was crucial in deriving the correct value for \( \csc \theta \) once \( \sin \theta \) was known.