Problem 72 - Algebra and Trigonometry

Question

\(71-74\). Find all solutions of the equation that lie in the interval \([0, \pi]\) . State each answer correct to two decimal places. $$ \tan x=2 $$

Step-by-Step Solution

Answer

The solution is \( x = 1.11 \).

Step 1: Solve the Basic Trigonometric Equation

We begin by solving the equation \( \tan x = 2 \). To find \( x \), we take the arctangent of both sides: \( x = \tan^{-1}(2) \). Using a calculator, \( x \approx 1.1071 \) radians.

Step 2: Determine the General Solution for Tangent

The general solution for the tangent function is given by \( x = \tan^{-1}(2) + n\pi \), where \( n \) is any integer. This accounts for the periodic nature of the tangent function, which has a period of \( \pi \).

Step 3: Find Solutions within the Given Interval

We need solutions in the interval \([0, \pi]\). Substituting \( n = 0 \), we get \( x \approx 1.1071 \). Substituting \( n = 1 \), we get \( x \approx 1.1071 + \pi \approx 4.2487 \), which is beyond the interval. Thus, only \( x \approx 1.1071 \) is within \([0, \pi]\).

Step 4: Round the Solution to Two Decimal Places

Rounding the value found in the previous step to two decimal places, we find the solution \( x \approx 1.11 \).

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function that relates a right triangle's opposite side to its adjacent side in a given angle. Mathematically, it is expressed as \( \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \), where \( x \) is an angle in radians or degrees.
The tangent function has some unique properties:

• It is periodic, repeating every \( \pi \) radians (or 180 degrees).
• Unlike sine and cosine, tangent does not have a maximum or minimum limit; it ranges from \(-\infty\) to \(+\infty\).
• The function has vertical asymptotes at every odd multiple of \( \frac{\pi}{2} \), where it is undefined.

Understanding these properties helps when solving trigonometric equations, as it is crucial to account for tangent's periodicity and undefined points. This is particularly important as the tangent function can be mapped onto a unit circle, where its periodicity and symmetry can be visualized.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle when the trigonometric value is known. For tangent, this inverse is denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \).

If you know the value of \( \tan(x) \) and want to find \( x \), you use the inverse tangent function. It's like the reverse operation of tangent. For example, if \( \tan(x) = 2 \), then \( x = \tan^{-1}(2) \).
Inverse trigonometric functions have their domains and ranges:

• The domain of \( \tan^{-1}(x) \) is all real numbers \(( -\infty, \infty )\).
• The range for \( \tan^{-1}(x) \) is \(( -\frac{\pi}{2}, \frac{\pi}{2} )\). This means you will get an angle in this specific interval.

In problem-solving scenarios, using inverse trigonometric functions is essential for reversing the process and finding alternative solutions to trigonometric equations.

General Solution for Tangent

The general solution of a trigonometric equation accounts for the periodic nature of trigonometric functions. For the tangent function, this is necessary due to its periodicity.

When we solve \( \tan(x) = k \) and find \( x = \tan^{-1}(k) \), the general solution incorporates the periodicity of \( \pi \). It is expressed as \( x = \tan^{-1}(k) + n\pi \), where \( n \) is any integer.
This general form illustrates why multiple solutions exist, each differing by a full cycle of \( \pi \).
The concept of the general solution:

• Accounts for all possible angles \( x \) for which the equation holds true.
• Defines a complete set of solutions by utilizing tangent's periodicity.
• Is critical in scenarios where solutions must be within specified intervals.

Always verify which solutions fall within the desired interval, as seen in our exercise where we only picked solutions within \([0, \pi]\). This method ensures we correctly identify the valid solutions for trigonometric equations.