Problem 24 - Calculus

Question

Find every \(\theta\) that satisfies the equation. $$ \sec \theta=-2 $$

Step-by-Step Solution

Answer

\( \theta = 2\pi/3 + 2k\pi \) and \( \theta = 4\pi/3 + 2k\pi \), where \( k \) is an integer.

Step 1: Understand the Definition of Secant

The secant function is the reciprocal of the cosine function, i.e., \( \sec \theta = \frac{1}{\cos \theta} \). Thus, \( \sec \theta = -2 \) translates to \( \frac{1}{\cos \theta} = -2 \).

Step 2: Solve for Cosine

To find \( \cos \theta \), reciprocate both sides of the equation to get \( \cos \theta = -\frac{1}{2} \).

Step 3: Determine the Quadrants

Since the cosine is negative, \( \theta \) must be in the second or third quadrant, where cosine values are negative.

Step 4: Find Reference Angle

The reference angle corresponding to \( \cos \theta = \frac{1}{2} \) is \( \theta = \pi/3 \) or \( 60^{\circ} \). This is because \( \cos(\pi/3) = 1/2 \).

Step 5: Apply Quadrant Adjustments

For the second quadrant, \( \theta = \pi - \pi/3 = 2\pi/3 \). For the third quadrant, \( \theta = \pi + \pi/3 = 4\pi/3 \).

Step 6: General Solution

Since trigonometric functions are periodic, the general solution includes all coterminal angles: \[ \theta = 2\pi/3 + 2k\pi \quad \text{and} \quad \theta = 4\pi/3 + 2k\pi, \text{ where } k \text{ is any integer.}\]

Key Concepts

Secant Function

The secant function is an important trigonometric function denoted by \( \sec \theta \). It is defined as the reciprocal of the cosine function, which means \( \sec \theta = \frac{1}{\cos \theta} \). This relationship is crucial when solving trigonometric equations involving the secant function. For the equation \( \sec \theta = -2 \), we can rewrite it using the reciprocal identity as \( \frac{1}{\cos \theta} = -2 \). By doing so, it becomes easier to solve for \( \cos \theta \), leading to further insights into the values of \( \theta \). Understanding this reciprocal identity allows you to transform and solve equations that might seem challenging at first glance.

Cosine Function

The cosine function, \( \cos \theta \), is one of the primary trigonometric functions. It gives the ratio of the adjacent side to the hypotenuse in a right triangle. When encountering the equation \( \frac{1}{\cos \theta} = -2 \), we solve for \( \cos \theta \) by taking the reciprocal of both sides. This yields \( \cos \theta = -\frac{1}{2} \).

• The cosine value of \(-\frac{1}{2}\) is significant because it helps indicate in which quadrants \( \theta \) can lie, the basis for solving our trigonometric equation.
• This step crucially links the secant equation directly to a more commonly used cosine value in trigonometry.

Recognizing cosine values and their respective angles is a vital skill that will aid in solving many trigonometric equations.

Reference Angle

A reference angle is the acute angle that a terminal side of an angle makes with the horizontal axis. In trigonometry, understanding the reference angle helps us find solutions for angles across all quadrants. When we state that \( \cos \theta = -\frac{1}{2} \), we look for its reference angle. Known to many students is that the cosine of \( \frac{1}{2} \) corresponds to \( \theta = \frac{\pi}{3} \) or \(60^{\circ}\).

• This means, for \( \cos \theta = -\frac{1}{2} \), the reference angle remains \( \frac{\pi}{3} \), allowing us to find values in applicable quadrants where cosine is negative.
• It reflects a simplified approach to tackling angle problems by breaking them into manageable parts. Recognizing the reference angle promptly can simplify solving complex trigonometric equations.

Quadrant Analysis

Quadrant analysis is key in the study of trigonometric functions as it helps determine the sign of a trigonometric function's value. For the equation \( \cos \theta = -\frac{1}{2} \), it's essential to know in which quadrants cosine is negative. Cosine is always negative in the second and third quadrants.

• For the second quadrant, we adjust the reference angle \( \frac{\pi}{3} \) to find \( \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \).
• For the third quadrant, we use \( \pi + \frac{\pi}{3} = \frac{4\pi}{3} \).
• This quadrant knowledge ensures the accuracy of your answers in angle reduction and approximation, leading to the general solution by adding the period \(2\pi\).

Comprehending quadrants can be immensely useful in predicting the behavior and values of trigonometric functions across angles, which makes solving the equation clearer and more structured.