Problem 26 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

What are the solutions of the equation \(\tan x+1=0\) for \(0 \leq x<2 \pi\) ? (A) \(\frac{\pi}{4}\) (B) \(\frac{3 \pi}{4}\) (C) \(\frac{5 \pi}{4}\) (D) \(\frac{7 \pi}{4}\)

Step-by-Step Solution

Answer

The solutions of the equation \(\tan x + 1 = 0\) for \(0 \leq x < 2 \pi\) are \(\frac{3\pi}{4}\) and \(\frac{7\pi}{4}\), hence the selected options are (B) and (D).

Step 1: Isolate the Tangent Function

First, isolate the tangent function by rearranging the original equation. This can be done by subtracting 1 from each side of the equation. The result is \(\tan x= -1\).

Step 2: Calculate the Inverse Tangent

Next, to find the angle that satisfies the equation, calculate the inverse tangent of -1 using the notation \(\tan^{-1}(-1)\). This corresponds to the angle \(-\frac{\pi}{4}\). However, our solution must be in the domain from 0 to \(2\pi\), so we need to find the equivalent positive angle.

Step 3: Find Equivalent Angle

As the tangent function has a period of \(\pi\), adding \(\pi\) to the current angle will provide an equivalent angle in the range. So \(-\frac{\pi}{4} + \pi = \frac{3\pi}{4}\).

Step 4: Find All Solutions within the Given Range

Lastly, as the cycle of the tangent function repeats every \(\pi\), add \(\pi\) to the obtained value until it reaches \(2\pi\). So adding \(\pi\) to \(\frac{3\pi}{4}\) gives \(\frac{7\pi}{4}\), which is within our range of \(0 \leq x < 2 \pi\). Therefore, the solutions within this range for \(\tan x + 1 = 0\) are \(\frac{3\pi}{4}\) and \(\frac{7\pi}{4}\).

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions, often denoted as \(\tan x\). It's defined as the ratio of the sine and cosine functions, meaning \(\tan x = \frac{\sin x}{\cos x}\). This particular function is known for its periodic nature with a period of \(\pi\) radians. Unlike sine and cosine, which cycle every \(2\pi\), tangent repeats its values every \(\pi\), making it unique among the trigonometric functions.
Understanding this periodicity helps when solving equations involving \(\tan x\). For instance, the equation \(\tan x = -1\) implies finding the angles \(x\) where tangent equals -1. Knowing the period allows us to calculate further potential solutions by simply adding multiples of \(\pi\) to any initial solution within a specified range.

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial when solving trigonometric equations since they allow us to find angle measures from known trigonometric values. For the tangent function, the inverse is denoted as \(\tan^{-1} x\), also called "arctangent." This function essentially gives us the angle whose tangent is \(x\).
In the case of \(\tan x = -1\), solving for \(x\) involves calculating \(\tan^{-1}(-1)\). The result, \(-\frac{\pi}{4}\), however, needs adjustment to fit within the standard range of \([0, 2\pi)\). Utilizing the periodic nature of tangent, we add \(\pi\) until the angle conforms to our desired interval.

Angle Measures

Angle measures are an essential part of working with trigonometric functions. They are usually expressed in radians for mathematical applications. One complete revolution around a circle is defined as \(2\pi\) radians. When solving equations like \(\tan x + 1 = 0\), we seek specific angle measures within a given range, like \([0, 2\pi)\). Each angle measure provides a specific solution where our trigonometric equation holds true. This is why adjusting solutions (like converting \(-\frac{\pi}{4}\) to \(\frac{3\pi}{4}\)) is important—ensuring all solutions are within the specified range and applicable context.

Periodicity of Trigonometric Functions

The periodicity of trigonometric functions is central to understanding how they behave. This periodic nature means that the functions' values repeat at regular intervals. For tangent, this period is \(\pi\), meaning \(\tan x = \tan(x + n\pi)\) for any integer \(n\). Knowing this is invaluable when solving trigonometric equations. Once you find one solution, you can generate all other solutions by adding or subtracting full periods—specifically \(\pi\) in the case of tangent. For example, given \(\tan x = -1\) with an initial solution \(\frac{3\pi}{4}\), adding \(\pi\) yields the next solution \(\frac{7\pi}{4}\), within the desired range up to \(2\pi\). Recognizing this regular pattern simplifies finding comprehensive solutions.