Problem 108 - Algebra and Trigonometry

Question

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \tan x \sec x=2 \tan x $$

Step-by-Step Solution

Answer

The solutions to the equation \(\tan x \sec x=2 \tan x\) on the interval [0,2\(\pi\)) are \(x = \frac{\pi}{3}, \frac{5\pi}{3}\)

Step 1: Simplify by canceling common factors

First, cancel out the common factor \(\tan (x)\) on both sides of the equation, resulting in \(\sec (x) = 2\). It is important to realize that when simplifying the equation in this way, we’re assuming \(\tan (x) ≠ 0\), so x cannot equal any odd multiple of \(\frac{\pi}{2}\) (i.e., when \(\cos (x) = 0)\). We will have to check these cases separately.

Step 2: Rewrite in terms of cosine

We know that \(\sec (x)\) is the reciprocal of \(\cos (x)\), so the equation becomes \(\cos (x) = \frac{1}{2}\). To solve this equation on the interval \([0,2\pi)\), we look for the angles in the unit circle where the x-coordinate (cosine value) is \(\frac{1}{2}\). This occurs at \(x = \frac{\pi}{3}, \frac{5\pi}{3}\).

Step 3: Check excluded solutions from Step 1

Remember that x ≠ any odd multiple of \(\frac{\pi}{2}\) from Step 1. Meaning that x cannot be \(\frac{\pi}{2}, \frac{3\pi}{2}\). But, by plugging \(x = \frac{\pi}{2}\) or \(x = \frac{3\pi}{2}\) into the original equation, no true statement is derived, due to \(\tan (x)\) being undefined for these values as reasoned earlier. Hence there is no additional solutions.

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers along with their endpoints. It’s especially useful in calculus, algebra, and trigonometry to specify the range of possible solutions or values. For this exercise, we focus on the interval

• defines a closed interval from 0 to just under 2π. It is important to note that 0 is included, while 2π is not. Closed intervals are denoted by square brackets, indicating the endpoints are part of the interval. When dealing with open intervals, parentheses are used instead.

Learning how to work within a given interval helps in understanding where the solution is valid and prevents errors in calculations.

This interval is used in trigonometric problems to limit the solutions to one cycle on the unit circle. It ensures solutions are precise and applicable to the situation at hand.

Cosine Function

The cosine function is a fundamental trigonometric function that links the angle of a triangle to the ratio of the adjacent side over the hypotenuse in a right triangle. On the unit circle, the cosine of an angle is represented by the x-coordinate of the point.

• For angles where the cosine value is \(\frac{1}{2}\), the equation \(\cos(x) = \frac{1}{2}\) tells us to find points on the unit circle where this occurs.
• In the interval \([0, 2\pi)\), these points are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).

The cosine function repeats every \(2\pi\), which means these points occur at regular intervals, making it predictable and easy to find additional solutions in extended intervals.

Understanding the cosine function is crucial for solving equations involving trigonometric identities and manipulating expressions effectively.

Tangent Function

The tangent function relates to the sine and cosine functions. It is defined as the ratio of the sine to the cosine. This means that tangent can be represented as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

• The tangent function tends to become undefined where the cosine is 0, specifically at odd multiples of \(\frac{\pi}{2}\).
• In the exercise, this means points like \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) need special consideration because solutions cannot occur there.

The tangent function has a period of \(\pi\), meaning it repeats its values every \(\pi\) units. This can lead to many recurring solutions in broader intervals.

Being familiar with these properties can help in identifying permissible and impermissible solutions and generally understanding the behavior of tangent in different equations.

Secant Function

The secant function is closely related to the cosine function since it is simply the reciprocal. It is defined as \(\sec(x) = \frac{1}{\cos(x)}\).

• In our equation, setting \(\sec(x) = 2\) was a crucial step to translate the problem into a form \(\cos(x) = \frac{1}{2}\) that is easier to work with.
• The secant function takes on large values where the cosine is small. It becomes undefined when the cosine function equals zero.

Understanding secant is essential for solving equations where direct use of cosine isn't straightforward.

By recognizing how secant interacts with other trigonometric functions, you can break down complex problems into simpler parts, often leading to solutions that are quickly validated on the unit circle.