Question

what is the exact value of
1-2sin(15degrees)?

also

2(sin2x)=2sinx solve it for 0<_x_<360

and also 2cos2x=-ã3 solve it and find points of intersection for 0<_x_<ƒÎ

Answer 1

Well, 2sin15 is the same as sin30, right? and sin30 has a special value of 1/2...1-1/2=1/2...so the exact value of 1-2sin15 is 1/2.

I would answer your second question, but the characters are all messed up and I can't understand it!

Answer 2

2sin15 is NOT equal to sin30

let's use the identity
cos 2A = 1 - 2sin^2 A
cos 30 = 1 - 2sin^2 15
√3/2 = 1 - 2sin^2 15
sin^2 15 = 1 - √3/2
= (2-√3)/4
sin 15 = √[(2-√3)/4]

now you can form 1 - 2sin 15,
I will let you finish it.

for the second,
2(sin2x)=2sinx
sin 2x = sinx
2sinxcosx - sinx = 0
sinx(2cosx - 1) = 0
sinx = 0 ----> x = 0 or x= 180 or x=360

or

cosx = 1/2 --- x = 60 or x = 300

the third is easy,
cos 2x = √3/2
2x = 30 or 2x = 330
so x = 15 or x = 165
since the period of cos 2x is 180, add 180 to 15 and 165 to get two more answers