SOLUTION: please solve this problem: please solve the trigonometric equation {{{ cos(2x)}}} - {{{sIn^2x/2 }}} + {{{3/4}}} = {{{0}}} give all positive values of the angle between 0 to 3

Algebra ->  Trigonometry-basics -> SOLUTION: please solve this problem: please solve the trigonometric equation {{{ cos(2x)}}} - {{{sIn^2x/2 }}} + {{{3/4}}} = {{{0}}} give all positive values of the angle between 0 to 3      Log On


   

Question 86882: please solve this problem: please solve the trigonometric equation
+cos%282x%29 - sIn%5E2x%2F2+ + 3%2F4 = 0
give all positive values of the angle between 0 to 360 degress give any approximate value to the nearest minute only.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

please solve this problem: please solve the trigonometric equation 

+cos%282x%29 - sIn%5E2x%2F2+ + 3%2F4 = 0 

give all positive values of the angle between 0 to 360 degress 
give any approximate value to the nearest minute only. 

+cos%282x%29 - sIn%5E2x%2F2+ + 3%2F4 = 0


Multiply the equation through by LCD = 4

4%2Acos%282x%29 - 4%2AsIn%5E2x%2F2+ + 4%2A3%2F4 = 4%2A0 

Simplifying:

4cos%282x%29 - 2sIn%5E2x+ + 3 = 0

Use the identity cos 2q = 1 - 2sin²q

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

4%281-2sIn%5E2x%29 - 2sIn%5E2x+ + 3 = 0

Remove the parentheses by distributing:

4-8sIn%5E2x%29 - 2sIn%5E2x+ + 3 = 0

Combine like terms:

7+-+10sIn%5E2x = 0

-10sIn%5E2x = -7

sIn%5E2x = %28-7%29%2F%28-10%29

sIn%5E2x = 0.7

Take the square roots of both sides:

sin%28x%29 = ±sqrt%280.7%29

sin%28x%29 = ±.8366600265

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

sIn%5E%28-1%29%28.8366600265%29 = 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%28x%29 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