Answer to Question #147769 in Trigonometry for Moustafa Elemam

Question #147769
(21 - 16〖sin〗^2 x+8 cosx)/(16 〖cos〗^2 x-29-8 √15 sin⁡x ) = 2

find x
1
Expert's answer
2020-12-01T06:23:09-0500
2116sin2x+8cosx16cos2x29815sinx=2\dfrac{21-16\sin^2x+8\cos x}{16\cos^2x-29-8\sqrt{15}\sin x}=2

The Pythagorean Identity


sin2x+cos2x=1\sin^2x+\cos^2x=1


Then


5+16(1sin2x)+8cosx16(cos2x1)13815sinx=2\dfrac{5+16(1-\sin^2x)+8\cos x}{16(\cos^2x-1)-13-8\sqrt{15}\sin x}=2

16cos2x+8cosx+516sin2x815sinx13=2\dfrac{16\cos^2x+8\cos x+5}{-16\sin^2x-8\sqrt{15}\sin x-13}=2

Let u=sinx,v=cosx.u=\sin x, v=\cos x. Then


{u2+v2=116v2+8v+5=32u21615u26\begin{cases} u^2+v^2=1 \\ 16v^2+8v+5=-32u^2-16\sqrt{15}u -26 \end{cases}




{u2+v2=1(16v2+8v+1)+2(16u2+815u+15)=0\begin{cases} u^2+v^2=1 \\ (16v^2+8v+1)+2(16u^2+8\sqrt{15}u +15)=0 \end{cases}


{u2+v2=1(4v+1)2+2(4u+15)2=0\begin{cases} u^2+v^2=1 \\ (4v+1)^2+2(4u+\sqrt{15})^2=0 \end{cases}

Then

{u2+v2=14v+1=04u+15=0\begin{cases} u^2+v^2=1 \\ 4v+1=0 \\ 4u+\sqrt{15}=0 \end{cases}

Check


16sin2x815sinx130-16\sin^2x-8\sqrt{15}\sin x-13\not=0

16u2815u130-16u^2-8\sqrt{15}u-13\not=0

u=154:u=-\dfrac{\sqrt{15}}{4}:


16(154)2815(154)13-16(-\dfrac{\sqrt{15}}{4})^2-8\sqrt{15}(-\dfrac{\sqrt{15}}{4})-13

=15+3013=20=-15+30-13=2\not=0

Hence

cosx=14\cos x=-\dfrac{1}{4}

x=±(πarccos(14))+2πn,nZx=\pm(\pi-\arccos(\dfrac{1}{4}))+2\pi n, n\in\Z


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