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marin [14]
3 years ago
15

Find all solutions in the interval [0, 2π). cos2x + 2 cos x + 1 = 0

Mathematics
1 answer:
iragen [17]3 years ago
4 0

Answer:

x = pi/2, 3pi/2            x = pi

Step-by-step explanation:

cos2x + 2 cos x + 1 = 0

           cos 2x = cos^2 - sin^2x  trig identity

cos^2x -sin^2 + 2cos x +1 = 0

rearrange

cos^2 x +2cos x + 1-sin^2 x = 0

       1 - sin^2 x = cos^2 x  trig identity

cos^2 x + 2 cos x + cos ^2 x = 0

combine like terms

2 cos ^2 x + 2 cos x = 0

divide by 2

cos ^2 x + cos x = 0

factor out a cos x

cos (x) (cos x +1) = 0

using the zero product property

cos (x) = 0       cos x + 1 =0

cos x =0         cos x = -1

taking the arccos of each side

arccos cos x = arccos (0)   arccos (cos x) = arccos (-1)

x = pi/2, 3pi/2            x = pi



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Y = (x)^(1/9)*Find f(x) when x=(1/2)Round your answer to the nearest thousandth.
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\times\text{ = }\frac{1}{2}

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y\text{ = f ( }\frac{1}{2}\text{ ) = (}\frac{1}{2})^{\frac{1}{9}}

We will apply the power on both numerator and denominator as follows:

f(\frac{1}{2})=\frac{1^{\frac{1}{9}}}{2^{\frac{1}{9}}}\text{ = }\frac{1}{2^{\frac{1}{9}}}

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f\text{ ( }\frac{1}{2})\text{ = }0.92587

Once, we have evaluated the answer in decimal form ( 5 decimal places ). We will round off the answer to nearest thousandths.

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However, since the (4th decimal place) = 8 > 5. Then we add ( 1 ) to the 3rd decimal place and truncate the rest of the decimal places i.e ( 4th and onwards ).

The answer to f ( 1 / 2 ) to the nearest thousandth would be:

\textcolor{#FF7968}{f}\text{\textcolor{#FF7968}{ ( }}\textcolor{#FF7968}{\frac{1}{2})}\text{\textcolor{#FF7968}{ = 0.926}}

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