Question 86949
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Please help me solve equation 

{{{ cos(2x)}}} - {{{sIn^2x/2 }}} + {{{3/4}}} = {{{0}}}

please give values between 0 to 360 degrees and give 
values to the nearest minute.
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Multiply the equation through by LCD = 4

{{{4*cos(2x)}}} - {{{4*sIn^2x/2 }}} + {{{4*3/4}}} = {{{4*0}}} 

Simplifying:

{{{4cos(2x)}}} - {{{2sIn^2x }}} + {{{3}}} = {{{0}}}

Use the identity cos 2<font face = "symbol">q</font> = 1 - 2sin²<font face = "symbol">q</font>

to substitute 1 - 2sin²x for cos(2x) in the first term:

{{{4(1-2sIn^2x)}}} - {{{2sIn^2x }}} + {{{3}}} = {{{0}}}

Remove the parentheses by distributing:

{{{4-8sIn^2x)}}} - {{{2sIn^2x }}} + {{{3}}} = {{{0}}}

Combine like terms:

{{{7 - 10sIn^2x}}} = {{{0}}}

{{{-10sIn^2x}}} = {{{-7}}}

{{{sIn^2x}}} = {{{(-7)/(-10)}}}

{{{sIn^2x}}} = {{{0.7}}}

Take the square roots of both sides:

{{{sin(x)}}} = ±{{{sqrt(0.7)}}}

{{{sin(x)}}} = ±{{{.8366600265}}}

Find the inverse sine of .836600265 in the
first quadrant:

{{{sIn^(-1)}}}{{{(.8366600265)}}} = 56.78908924°

To change the decimal part of that to minutes, 
multiply the decimal part .78908924 by 60, getting
47.34535435' then round to the nearest minute, so
the value of x is the first quadrant is 56°47'.

But since the {{{sin(x)}}} can be positive or 
negative, we will get all the angles in all the
quadrants which have 56²47 as their reference
angles.

The second quadrant answer is found by
subtracting 56°47' from 180° or

180° - 56°47' = 179°60' - 56°47' = 123°13'

The third quadrant answer is found by
adding 56°47' to 180° or

180° + 56°47' = 236°47'

The fourth quadrant answer is found by
subtracting 56°47' from 360°

360° - 56°47' = 359°60' - 56°47' = 303°13'

So all the answers for x between 0° and 360° are

x = 56°47', 123°13', 236°47', and 303°13

Edwin</pre>