Problem 103 - Precalculus Essentials

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. $$\tan x=-4.7143$$

Step-by-Step Solution

Answer

The solutions to the equation \( \tan x = -4.7143 \) can be obtained by finding the reference angle and then calculating the angles for the specific quadrants. Make sure to verify that these solutions indeed satisfy the original equation and are in the given interval.

Step 1: Analyze the Quadrants

First, it's critical to identify in which quadrants tangent is negative. Tangent is negative in quadrants II and IV since tangent is the ratio of the sine to the cosine, and sine is negative in quadrants III and IV, while cosine is negative in quadrants II and III.

Step 2: Determine the Reference Angle

Next, the reference angle, which is the angle formed by the ray and the x-axis, should be found. Since the tangent is negative, we ignore the sign when finding the reference angle. Hence, we calculate the reference angle \( \theta = \arctan |\frac{-4.7143}{1}| \). To get an accurate answer, we use approximate values instead of exact values. This gives us \( \theta \) in the first quadrant.

Step 3: Find the Solutions

We now determine the actual solutions, which are the angles in the second and fourth quadrant. The angle in the second quadrant is given by \( \pi - \theta \) and the angle in the fourth quadrant is given by \( 2\pi - \theta \).

Step 4: Verify the Solutions

The last step is to verify that the computed angles are solutions to the original equation and fall in the interval from 0 to \( 2\pi \).

Key Concepts

Understanding the Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine of an angle to its cosine, which can be expressed mathematically as: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]This function is periodic, which means it repeats its values at regular intervals. The period of the tangent function is \( \pi \), whereas the sine and cosine functions have a period of \( 2\pi \). This unique quality causes the tangent function to have vertical asymptotes wherever the cosine function is zero since division by zero is undefined.

• The tangent function is positive in the first and third quadrants where sine and cosine have the same sign.
• It switches to negative in the second and fourth quadrants, introducing interesting behavior when solving trigonometric equations.

When addressing equations like \( \tan x = -4.7143 \), the negative value indicates a solution where tangent values are negative, specifically in the second and fourth quadrants.

What is a Reference Angle?

Reference angles are a handy concept when working with trigonometric functions. They simplify the calculation of trigonometric function values by relating them to acute angles within the first quadrant.

To find the reference angle \( \theta_R \), consider these steps:

• Always take the absolute value of the original tangent equation (i.e., ignore the negative sign), as the reference angle is always positive.
• Calculate \( \theta_R \) as \( \tan^{-1} |\text{value}| \). This will give you an acute angle in the first quadrant.

For an equation \( \tan x = -4.7143 \), ignoring the negative sign helps us to identify the acute angle that forms the basis for determining exact solutions in other quadrants.

Understanding the use of a reference angle is crucial as it helps you find the actual angle in the specified quadrants where the function takes the specific sign (positive or negative) as needed.

Determining Quadrants for Solutions

In trigonometry, quadrants play a major role in analyzing angles and solving equations. There are four quadrants on the coordinate plane, each with distinct characteristics regarding the signs of trigonometric functions. Understanding these helps us determine where tangent or any trigonometric function can be positive or negative.

For the tangent function:

• Quadrant I: \( \tan \) is positive because both \( \sin \) and \( \cos \) are positive.
• Quadrant II: \( \tan \) is negative because \( \sin \) is positive while \( \cos \) is negative.
• Quadrant III: \( \tan \) is positive again since both \( \sin \) and \( \cos \) are negative.
• Quadrant IV: \( \tan \) is negative as \( \sin \) is negative and \( \cos \) is positive.

When solving equations like \( \tan x = -4.7143 \), solutions are found by placing the reference angle in the quadrants where the tangent is negative, specifically quadrants II and IV.
You derive the total angle for each quadrant by formulas \( \pi - \theta_R \) for quadrant II and \( 2\pi - \theta_R \) for quadrant IV. This method ensures solutions fall within specified boundaries, providing the correct answers for the given equation.