How do I solve (2sqrt2-2)cosalpha=3sin^2alpha+sqrt2-4?

3 Answers
Jun 10, 2018

:. alpha=2kpi+-arccos[{-(sqrt2-1)+-root(4)2}/3], k in ZZ.

Explanation:

Replacing sin^2alpha in the right member by (1-cos^2alpha), we have,

(2sqrt2-2)cosalpha=3-3cos^2alpha+sqrt2-4.

:. 3cos^2alpha+2(sqrt2-1)cosalpha-(sqrt2-1)=0.

This is a quadr. eqn. in cosalpha, and, its discriminant

Delta=(2sqrt2-2)^2+4*3*(sqrt2-1),

=(8-8sqrt2+4)+12sqrt2-12,

:. Delta=4sqrt2.

Therefore, using the quadr. formula, we have,

cosalpha={-2(sqrt2-1)+-sqrt(4sqrt2)}/(2*3),

={-2(sqrt2-1)+-2root(4)2}/(2*3),

:. cosalpha={-(sqrt2-1)+-root(4)2}/3.

:. alpha=2kpi+-arccos[{-(sqrt2-1)+-root(4)2}/3], k in ZZ.

Jun 10, 2018

=>alpha=2npi+-cos^-1((1-sqrt2+2^(1/4))/3) or
alpha=2npi+-cos^-1((1-sqrt2-2^(1/4))/3)

Explanation:

" "(2sqrt2-2)cosalpha=3sin^2alpha+sqrt2-4
=>2(sqrt2-1)cosalpha=3(1-cos^2alpha)+sqrt2-4

let cosalpha=y", then equation becomes"
2(sqrt2-1)y=3(1-y^2)+sqrt2-4
=>3y^2+2(sqrt2-1)y-3-sqrt2+4=0
=>3y^2+2(sqrt2-1)y+(1-sqrt2)=0

now solve for y, from quadratic formula
y=(-2(sqrt2-1)+-sqrt((2(sqrt2-1))^2-4(3)(1-sqrt2)))/(2(3))
=(2(1-sqrt2)+-sqrt(4(2+1-2sqrt2)-12+12sqrt2))/6
=(2-2sqrt2+-2sqrt(sqrt2))/6
=>cosalpha=(2-2sqrt2+-2sqrt(sqrt2))/6
=> cosalpha=(1-sqrt2+2^(1/4))/3" or " cosalpha=(1-sqrt2-2^(1/4))/3
=>alpha=cos^-1((1-sqrt2+2^(1/4))/3) or
alpha=cos^-1((1-sqrt2-2^(1/4))/3)
since cosx repeats itself after 2npi period (ninZZ) " add " 2npi for general solutions

Jun 10, 2018

x = +- 74^@93 + k360^@
x = +- 122^@68 + k360^@

Explanation:

(2sqrt2 - 2)cos x = 3sin^2 x + sqrt2 - 4
(2sqrt2 - 2)cos x = 3 - 3cos^2 x + sqrt2 - 4
3cos^2 x + 2(sqr2 - 1))cos x - (sqrt2 - 1) = 0
3cos^2 x + 0.828cos x - 0.414 = 0
Solve this quadratic equation for cos x.
D = d^2 - 4ac = 0.686 + 4.968 = 5.65 --> d = +- 2.38
There are 2 real roots:
cos x = -b/(2a) +- d/(2a) = - 0.828/6 +- 2.38/6 = - 0.138 +- 0.40
cos x = - 0.54, and cos x = 0.26
a. cos x = - 0.54
Calculator and unit circle give 2 solutions:
x = +- 122^@68 + k360^@
b. cos x = 0.26 -->
x = +- 74^@93 + k360
Check by calculator:
1. x = 74.93 --> 0.83cos x = 0.22 --> (2sqr2 - 2)cos x = 0.22
3sin^2 x + sqrt2 - 4 = 2.8 + 1,41 - 4 = 0.21. Proved
2. x = 122.68 --> (2sqrt2 - 2)cos x = - 0.45
3sin^2 x + sqrt2 - 4 = 2.13 + 1.42 - 4 = - 0.45. Proved