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Sagot :
Answer:
all values of x between 0 and 2 pi are 0.12, 1.45, 3.24, and 4.59.
In degrees, this is 6.88, 83.1, 185.6 and 262.98 degrees.
Explanation:
First we realize that
[tex]\sin 2x=2\sin x\cos x[/tex]therefore, the equation
[tex](2\sin x\cos x)^2+4\sin 2x=1[/tex]becomes
[tex](\sin 2x)^2+4\sin 2x=1[/tex]We can complete the square here by adding 4 to both sides. This gives
[tex](\sin 2x)^2+4\sin 2x+4=5[/tex][tex](\sin 2x+2)^2=5[/tex]taking the square root of both sides gives
[tex]\sin 2x+2=\sqrt[]{5}[/tex]Subtracting 2 from both sides gives
[tex]\sin 2x=\sqrt[]{5}-2[/tex]Finally, taking the inverse of both sides gives
[tex]2x=\sin ^{-1}\lbrack\sqrt[]{5}-2\rbrack[/tex][tex]x=\frac{\sin ^{-1}\lbrack\sqrt[]{5}-2\rbrack}{2}[/tex]The first two values of the inverse function are
[tex]\begin{gathered} \frac{\sin ^{-1}\lbrack\sqrt[]{5}-2\rbrack}{2}=0.12 \\ \end{gathered}[/tex][tex]\frac{\sin^{-1}\lbrack\sqrt[]{5}-2\rbrack}{2}=1.45[/tex][tex]\frac{\sin^{-1}\lbrack\sqrt[]{5}-2\rbrack}{2}=3.24[/tex][tex]\frac{\sin^{-1}\lbrack\sqrt[]{5}-2\rbrack}{2}=4.59[/tex]Hence, all values of x between 0 and 2 pi are 0.12, 1.45, 3.24, and 4.59.
In degrees, this is 6.88, 83.1, 185.6 and 262.98 degrees.
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