Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \tan x+1=-1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\).

Step 1: Rearrange the equation

First, we start by handling the equation given: \(2 \tan x + 1 = -1\). Subtract 1 from both sides to isolate terms involving \(\tan x\): \(2 \tan x = -2\).

Step 2: Solve for \(\tan x\)

Now, divide both sides of the equation by 2 to solve for \(\tan x\): \(\tan x = -1\).

Step 3: Determine solutions for \(\tan x = -1\)

Now, we need to find values of \(x\) such that \(\tan x = -1\) within the given interval \([0, 2\pi)\). The tangent function is negative in the second quadrant, and also the result \(\frac{3\pi}{4}\) and in the fourth quadrant \(\frac{7\pi}{4}\). So the solutions are \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\).

Key Concepts

Tangent Function

Understanding the tangent function is crucial when solving trigonometric equations. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. In symbols, \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). This function is related to the sine and cosine functions, expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

One important characteristic of the tangent function is its periodicity. The function repeats its values every \( \pi \, \text{radians} \). This means if \( \tan \theta = a \), then \( \tan (\theta + n\pi) = a \), where n is any integer. Also, because the tangent function has values ranging from \(-\infty \) to \(+\infty \), unlike sine and cosine, it does not have maximum or minimum bounds but does have vertical asymptotes where \(\cos \theta = 0 \).

• Tangent is positive in the first and third quadrants.
• It is negative in the second and fourth quadrants.
• This helps to find solutions over specified intervals.

Unit Circle

The unit circle is a pivotal concept in understanding trigonometric functions. It is a circle of radius one unit centered at the origin of the coordinate plane. All points on the unit circle satisfy the equation \( x^2 + y^2 = 1 \).

On the unit circle, the angle \(\theta\) (measured in radians) corresponds to a point \((x, y)\), where \(x = \cos \theta\) and \(y = \sin \theta\). The tangent of the angle can be found using the coordinates, since \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}\).

The unit circle allows us to determine the sign of the tangent in different quadrants:

• Second quadrant: \(\tan \theta\) is negative because \(y/x = \text{negative}\).
• Fourth quadrant: \(\tan \theta\) is also negative.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities are fundamental tools in simplifying expressions and solving equations. Here are a few key identities that are often useful:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Tangent Identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Reciprocal Identities which relate to secant, cosecant, and cotangent.

Identities help to solve trigonometric equations without a calculator. Knowing which identities to apply can simplify the problem-solving process significantly. For example, when given an expression like \(2 \tan x = -2\), using the tangent identity can directly lead to solutions for \(x\) by manipulating the equation and understanding the specific properties of the tangent function.