Samuel started the 30-day challenge on the same day that Greg did he could do 18 push-ups on the first day and he set a goal to do two additional push-ups every four days write the equation that describes the relationship for the number of push-ups samuel does, f(x) in terms of x days

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samantha9430

22.10.2022

Mathematics

$\large\begin{array}{l} \textsf{Solve the trigonometric equation:}\\\\ \mathsf{3\,tan(2x+15^\circ)=4}\qquad\textsf{for }\mathsf{0^\circ \le x \le 180^\circ}\\\\ \mathsf{tan(2x+15^\circ)=\dfrac{4}{3}} \end{array}$

$\large\begin{array}{l} \textsf{Take the inverse tangent function of both sides:}\\\\ \mathsf{2x+15^\circ=tan^{-1}\!\left(\dfrac{4}{3}\right)+k\cdot 180^\circ}\\\\ \mathsf{2x=tan^{-1}\!\left(\dfrac{4}{3}\right)-15^\circ+k\cdot 180^\circ}\\\\ \mathsf{x=\dfrac{1}{2}\cdot \left[tan^{-1}\!\left(\dfrac{4}{3}\right)-15^\circ+k\cdot 180^\circ \right]}\\\\ \mathsf{x=\dfrac{1}{2}\,tan^{-1}\!\left(\dfrac{4}{3}\right)-\dfrac{15^\circ}{2}+k\cdot \dfrac{180^\circ}{2}}\end{array}$

$\large\begin{array}{l} \mathsf{x=\dfrac{1}{2}\,tan^{-1}\!\left(\dfrac{4}{3}\right)-7.5^\circ+k\cdot 90^\circ}\qquad\textsf{(exact form)}\\\\ \textsf{Computing the arctangent of }\mathsf{\dfrac{4}{3},}\textsf{ we get (approximatelly)}\\\\ \mathsf{tan^{-1}\!\left(\dfrac{4}{3}\right)\approx 53.13^\circ} \end{array}$

$\large\begin{array}{l} \textsf{Then,}\\\\ \mathsf{x\approx \dfrac{1}{2}\cdot 53.13^\circ-7.5^\circ+k\cdot 90^\circ}\\\\ \mathsf{x\approx 26.57^\circ-7.5^\circ+k\cdot 90^\circ}\\\\ \boxed{\begin{array}{c}\mathsf{x\approx 19.07^\circ+k\cdot 90^\circ} \end{array}}\qquad\textsf{(approximate form)}\\\\\\ \textsf{where k is an integer.} \end{array}$

__________

$\large\begin{array}{l} \textsf{Finding the x values that satisfy}\\\\ \mathsf{0^\circ\le x\le 180^\circ}\\\\\\ \bullet~~\textsf{For }\mathsf{k=0:}\\\\ \boxed{\begin{array}{c}\mathsf{x \approx 19.07^\circ} \end{array}}\qquad\checkmark\\\\\\ \bullet~~\textsf{For }\mathsf{k=1:}\\\\ \mathsf{x \approx 19.07^\circ+90^\circ}\\\\ \boxed{\begin{array}{c}\mathsf{x \approx 109.07^\circ} \end{array}}\qquad\checkmark \end{array}$

$\large\begin{array}{l} \textsf{Solution set: }\mathsf{S=\left\{19.07^\circ,\,109.07^\circ\right\}} \end{array}$

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$\large\textsf{I hope this helps. :-)}$

Tags: trigonometric equation trig tangent tan inverse tangent arctangent arctan

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