use a graphing utility to approximate the solutions of the equation in the interval 0, 2π) by collecting all…

use a graphing utility to approximate the solutions of the equation in the interval 0, 2π) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the x - intercepts of the graph. (round your answers to four decimal places. enter your answers as a comma - separated list. if there is no solution, enter no solution.) 5 sin(x)=cos(x) - 4

use a graphing utility to approximate the solutions of the equation in the interval 0, 2π) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the x - intercepts of the graph. (round your answers to four decimal places. enter your answers as a comma - separated list. if there is no solution, enter no solution.) 5 sin(x)=cos(x) - 4

Answer

Explanation:

Step1: Rearrange the equation

Collect all terms on one - side: $5\sin(x)-\cos(x)+4 = 0$.

Step2: Use a graphing utility

Graph the function $y = 5\sin(x)-\cos(x)+4$ on the interval $[0,2\pi]$. Then use the zero - or root - finding feature of the graphing utility. The general form of $a\sin(x)+b\cos(x)$ can be written as $A\sin(x +\varphi)$, where $A=\sqrt{a^{2}+b^{2}}$ and $\tan\varphi=\frac{b}{a}$. Here $a = 5$ and $b=-1$, so $A=\sqrt{5^{2}+(-1)^{2}}=\sqrt{25 + 1}=\sqrt{26}\approx5.0990$ and $\tan\varphi=-\frac{1}{5}$, $\varphi\approx - 0.1974$. The function is $y=\sqrt{26}\sin(x-\varphi)+4$. Using a graphing utility (such as a graphing calculator or software like Desmos), we find the $x$ - intercepts in the interval $[0,2\pi]$. The solutions are $x\approx1.9823,5.4265$.

Answer:

$1.9823,5.4265$