Question

In Exercises \(31-36,\) solve the equation in the specified interval. $$\sec x=-3\( \)-\pi\( \)\leq x<\( \)\pi$$

Short answer

The solutions to the equation within the interval \(-\pi \leq x < \pi\) are \(x = \pi - \cos^{-1}(1/3)\) and \(x = -\pi + \cos^{-1}(1/3)\).

Step-by-step solution

Convert secant to cosine

Rewrite the equation in terms of cosine. The secant of \(x\) can be written as \(1/\cos x\), so our equation becomes \(\cos x = -1/3\).

Use inverse cosine function

With the equation in terms of cosine now, the next step is using the arccos function, the inverse of cosine, to solve for \(x\). We can do this by taking \(\cos^{-1}(-1/3)\), which will give us the reference angle.

Determine reference angle

Due to the range of the cosine function, we find that the reference angle is not within the interval \(-\pi \leq x < \pi\). The cosine function is negative in both the second and the third quadrants, and the reference angle is likewise negative. However, the arccos function only returns values in the interval \(0 \leq x \leq \pi\).

Calculate x values

Knowing that the cosine function has period 2\(\pi\), and is negative in the second and third quadrants, we need to add and subtract \(\pi\) to our reference angle to find the values of \(x\) that satisfy the equation in the given range. Adding \(\pi\) gives a value in the third quadrant, while subtracting \(\pi\) gives a value in the second quadrant.

Key concepts

Inverse Trigonometric Functions

When solving trigonometric equations, understanding inverse trigonometric functions is paramount. These functions, often referred to as arcfunctions, allow us to work backwards from a known trigonometric ratio to find the corresponding angle. Inverse functions for sine, cosine, and tangent exist, commonly denoted as \(\arcsin\), \(\arccos\), and \(\arctan\) respectively.

For instance, if we know the cosine of an angle \(x\) is \( -1/3 \), using the inverse cosine function, denoted as \(\cos^{-1}\) or \(\arccos\), allows us to find the value of \(x\). We would express this as \( x = \arccos(-1/3) \). However, it's crucial to remember that the \(\arccos\) function only returns values within the range of \(0 \leq x \leq \pi\), which is from 0 to 180 degrees. This limitation means that \(\arccos\) will give us the principal value of the angle, which in the case of a negative cosine value, corresponds to an angle in the second quadrant.

Reference Angles in Trigonometry

Reference angles are a concept in trigonometry used to simplify the process of finding the exact values of trigonometric functions for any given angle. Importantly, a reference angle is always acute, meaning it is between \(0\) and \(\pi/2\) radians or \(0\) and \(90\) degrees. It is the smallest angle that the terminal side of the given angle makes with the x-axis.

Regardless of which quadrant an angle lies in, its trigonometric functions can be determined using its reference angle, keeping in mind the signs of the functions in that particular quadrant. For instance, cosine and secant are negative in the second and third quadrants, so when we solve for an angle like in the given exercise, we use the reference angle to determine the specific angles in the interval \( -\pi \leq x < \pi \) that possess a cosine value of \( -1/3 \). This approach simplifies the problem and makes finding the solution more manageable.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and it relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In the context of a unit circle or the coordinate plane, it represents the x-coordinate of a point on the circle corresponding to a given angle.

Cosine has certain properties that are essential when solving trigonometric equations. It has a period of \(2\pi\), meaning the function's values repeat every \(2\pi\) radians. Additionally, it is an even function, indicating that \(\cos(x) = \cos(-x)\). These characteristics are vital when determining all possible solutions to an equation within a specified interval.

In the exercise at hand, the negative cosine value suggests that the angle we are looking for lies either in the second or third quadrant, where the cosine value is negative. When we incorporate the periodicity of cosine function, this information aids us not only in finding the principal value but also in deducing all the possible angles that would satisfy the original equation within the given range.