91Ó°ÊÓ

The tangent function, often written as \( \tan x \), is a fundamental concept in trigonometry. It relates the sine and cosine functions using the formula \( \tan x = \frac{\sin x}{\cos x} \). This function can be utilized to simplify equations involving \( \sin x \) and \( \cos x \) into a single expression involving \( \tan x \).
In the provided exercise, the equation \( 3 \sin x = 7 \cos x \) is transformed by dividing both sides by \( \cos x \), resulting in \( 3 \tan x = 7 \). This step is crucial as it turns a trigonometric equation with two different trigonometric functions into one involving only \( \tan x \).
This simplification allows us to use the properties of the tangent function to find solutions to the equation. The \( \tan x \) function is periodic with a period of \( \pi \), meaning \( \tan(x + n\pi) = \tan x \) for any integer \( n \). This periodicity assists greatly in solving trigonometric equations.

The general solution of a trigonometric equation is an expression that includes all possible angle measures that satisfy the equation. For equations involving the tangent function, like \( \tan x = \frac{7}{3} \), we employ the arctangent function, \( \arctan \), to find the initial solution.
Here, we have \( x = \arctan\left(\frac{7}{3}\right) \). Given that \( \tan x \) is periodic with period \( \pi \), the general solution is expressed as \( x = \arctan\left(\frac{7}{3}\right) + n\pi \), where \( n \) is any integer.
This formulation accounts for all angles \( x \) that will result in \( \tan x\) equaling \( \frac{7}{3} \) because every \( \pi \) rotation will bring the tangent value back to the same point. When solving trigonometric equations, writing the general solution is a way to capture this periodicity and ensure no solution is missed.

Interval solutions are specific solutions to a trigonometric equation that fall within a given range. For the task at hand, solutions are sought within the interval \([0, 2\pi)\). This involves finding specific values of \( x \) from the general solution that are within this interval.
By calculating \( \arctan\left(\frac{7}{3}\right) \), we get \( x \approx 1.16590454 \) as our starting point. Next, we add \( \pi \) to this solution to find other solutions within the interval. Adding \( \pi \) gives approximately \( x \approx 4.30749768 \).
Both these values are within the given interval \([0, 2\pi)\). For trigonometric equations, especially when the solution involves \( \tan x \), it's common to check multiple cycles of \( \pi \) to ensure all solutions in the range are captured. It's also important to use a calculator for precision, particularly when working with approximations, like the ones required to five decimal places here.