Question 477525: Solve:
cos-¹ x + cos-¹ 2x = 60°
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Solve:
cos-¹ x + cos-¹ 2x = 60°
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cos A=x
sin A=√(1-x^2)
cos B=2x
sin B=√(1-4x^2)
Using Trig Identity: cos(A+B)=cos A*cos B-Sin A*Sin B
cos(A+B)=x*2x-(√(1-x^2)*√(1-4x^2)=cos 60º=1/2
cos(A+B)=2x^2-(1-5x^2+4x^4)=1/2
2x^2-1+5x^2-4x^4=1/2
-1+7x^2-4x^4-1/2=0
4x^4-7x^2+3/2=0
8x^4-14x^2+3=0
let u=x^2
u^2=x^4
8u^2-14u+3=0
(4u-1)(2x-3)=0
u=1/4
or
u=3/2
..
1/4=x^2
x=±1/2
arccos (-1/2)+arccos (2*-1/2)=-60º+180≠60º (reject)
arccos (1/2)+arccos (2*1/2)=60º+arccos(1)=60º+0=60º (answer)
..
3/2=x^2
x=±√(3/2)=1.2247 (reject, (0 ≤ cosx ≤1))
..
ans:
x=1/2
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