Problem 47 - Calculus: One and Several Variables

Question

Find the solutions \(x\) that are in the interval [0.2 \(\pi\) ]. Express your answers in radians and use four decimal place accuracy. $$\tan x=6.7192$$.

Step-by-Step Solution

Answer

The solutions of \(x\) in the interval [0, \(2\pi\)] are approximately \(x = 1.4053\) and \(x = 4.5469\) radians.

Step 1: Principal Solution

The principal solution can be found by using the inverse tangent function. So, we have \(x = \tan^{-1}(6.7192)\). It's important to check the mode your calculator is set to. Here, we want to calculate this in radian mode.

Step 2: Convert to Decimal

After calculating, we get the result approximately equal to 1.4053 radian. So, the principal solution for \(x\) that we calculate is \(x = 1.4053\) radian.

Step 3: Find Additional Solutions

The original equation, \(\tan x = 6.7192\), is periodic with period \(\pi\). So, to find additional solutions, we will add multiples of \(\pi\) to the principal solution, until we are still in the interval [0, \(2\pi\)]. Thus, the additional solution in the interval will be \(x = 1.4053 + \pi\), which equals approximately 4.5469 radians.

Key Concepts

Inverse Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent have inverse functions that help us determine angles when given a function value. For the tangent function, its inverse is denoted as \( \tan^{-1} \) or arctan. This inverse function will give you the angle whose tangent value is a specific number.

• The inverse tangent is essential in solving trigonometric equations because it allows you to work backwards to find angles.
• When using an inverse trigonometric function, be sure your calculator is set to the correct angle mode (degrees or radians), typically radians in higher mathematics as it relates better to calculus and continuous functions.
• The principal value is typically considered on the range \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) for the tangent function. This means if you solve \( x = \tan^{-1}(6.7192) \), you find the smallest angle whose tangent is 6.7192 within this range.

Understanding inverse functions helps analyze and solve problems involving angles and periodicity in mathematical exercises.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius-to-arc length ratio. One complete circle is \( 2\pi \) radians.

• Radians provide a natural way of representing angles, especially when dealing with trigonometric functions and calculus.
• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \). Conversely, convert radians to degrees by multiplying by \( \frac{180}{\pi} \).
• In this exercise, angles are expressed in radians for precision and simplicity when calculating periodic solutions.

Angles in radians easily integrate into trigonometric equations, allowing seamless transition between function values and angle measures.

Periodic Functions

Periodic functions repeat values in regular intervals. For the tangent function, this interval is \( \pi \). This means if you know one solution for \( \tan x = c\), additional solutions can be found by adding multiples of \( \pi \).

• Tangent's periodicity reflects in the solutions for \( \tan x = 6.7192 \), where the principal solution is 1.4053 radians.
• To find more solutions within a given interval, you use the principle that \( \tan(x) = \tan(x + n\pi) \) for any integer \( n \).
• This characteristic helps in broadening solutions beyond the principal value by incorporating periodicity up to the set boundary, here from 0 to \( 2\pi \).

This feature of periodicity simplifies solving trigonometric equations over extensive domains, providing the same value at regular intervals.