Question #d574e

1 Answer

Answer:

#x=45^@#is the correct value of #x#
If at #x=26.56505118^@# then #tan x=1/2#

Explanation:

Solve the trigonometric equation

#1+sin^2 x=3 sin x cos x#

Let #cos x=sqrt(1-sin^2 x)" "#from Pythagorean Relation

#1+sin^2 x=3 sin x sqrt(1-sin^2 x)#

Square both sides of the equation

#1+sin^2 x=3 sin x sqrt(1-sin^2 x)#

#(1+sin^2 x)^2=(3 sin x sqrt(1-sin^2 x))^2#

#1+2*sin^2 x+sin^4 x=9*sin^2 x(1-sin^2 x)#

#1+2*sin^2 x+sin^4 x=9*sin^2 x-9*sin^4 x#

#10*sin^4 x-7*sin^2 x+1=0#

Quadratic Equation

Let #w^2=sin^4 x# and #w=sin^2 x# then we have

#10*sin^4 x-7*sin^2 x+1=0#

#10*w^2-7*w+1=0#

then solve for w

#w=(-b+-sqrt(b^2-4ac))/(2a)#

#w=(-(-7)+-sqrt((-7)^2-4(10)(1)))/(2(10))#

#w=(7+-sqrt(49-40))/(2(10))#

#w_1=(7+3)/20=1/2#
#w_2=(7-3)/20=1/5#

#sin^2 x=+-sqrt(1/2)# and

#x=45^@# which is the root of the equation

#x=-45^@# not a root
#x=+-26.565^@# not a root

God bless...I hope the explanation is useful.