need step by step help as i have final coming up and I have forgot lots and I cant find my notes.

Question

1) Find exact value using sum or difference identities or double angle identities

a) tan 105

b)cos 75

 

2) solve for x

a)2sin²x-3sinx-2=0 ,    0≤x<2Ω

b)cos²x = cotx sinx ,    0≤x<360°

Answer 1

a) 2sin²x - 3sinx - 2=0

pretend that sin(x)=x, then the above equation can be rewritten as

2x² - 3x - 2=0

 

(2x + 1)(x - 2)

 

plug sin(x) back into equation:

(2sin(x) + 1)(sin(x) -2)=0

 

For the case 2sin(x) + 1=0

 

sin(x)=-1/2

x=210 degrees (7/12 π), 330 degrees (11/12 π)  These angles are locates 30 degrees downwards in the direction of the y-axis.

 

for the case sin(x) - 2=0:

 

sin(x)=2, no solution exists because -1≤sin(x)≤1.

 

x={7/12 pi, 11/12 pi}

 

b) cos²x = cotx sinx

cotx=cos(x)/sin(x), so

 

cos2x = cotx sinx is also cos2x = (cosx/sinx) sinx

sin(x) terms cancel each other

 

cos²x=cosx

 

cos²(x) - cos(x)=0

 

factor out cos(x):

cos(x)(cos(x) - 1)=0

 

when cos(x)=0

 

x=90, 270 degrees.

 (positive and negative directions of the y axis)

 

when cos(x) -1=0

cos(x)=1

 

x=0, 360 degrees (positive x-axis)

 

answers:

x={0, 90, 270, 360}