Trigonometry
posted by Tiffany Enlow on .
4. Find the exact value for sin(x+y) if sinx=4/5 and cos y = 15/17. Angles x and y are in the fourth quadrant.
5. Find the exact value for cos 165degrees using the halfangle identity.
1. Solve: 2 cos^2x  3 cosx + 1 = 0 for 0 less than or equal to x <2pi.
2. Solve: 2 sinx  1 = 0 for 0degrees less than or equal to x <360degrees.
3. Solve: sin^2x = cos^2x for 0degrees is less than or equal to x < 360degrees.
4. Solve sinx  2sinx cosx = 0 for 0 is less than or equal to x < 2pi.

#4
sin(x+y) = sinx cosy + cosx siny
so we need the cosx and the siny
from sinx = 4/5 we know we are dealing with the 3,4,5 rightangled triangle, so in the fourth quadrant cosx = 3/5
from cosy = 15/17 we know we are dealing with the 8,15,17 rightangled triangle
so in the fourth quadrant, siny = 8/17
then
sin(x+y) = sinx cosy + cosx siny
= (4/5)(15/17) + (3/5)(8/17)
= 84/85 
cos 2A = 2cos^2 A  1
let A = 165, then 2A = 330
so let's find cos 330
cos(330)
= cos(36030)
= cos360 cos30 + sin360 sin30
= (1)(√3/2) + (0)(1/2) = √3/2
then √3/2 = 2cos^2 165  1
(√3 + 2)/4 = cos^2 165
cos 165 = √(√3 + 2))/2
(algebraically, our answer would have been ± , but I picked the negative answer since 165 is in the second quadrant, and the cosine is negative in the second quadrant) 
I will give you hints for the rest,
you do them, and let me know what you get.
#1 factor it as
(2cosx  1)(cosx 1) = 0
so cosx  1/2 or cosx = 1
take it from there, you should get 3 answers.
#2, the easiest one
take the 1 to the other side, then divide by 2,
#3, take √ of both sides to get
sin 2x = ± cos 2x
sin2x/cos2x = ± 1
tan 2x = ± 1 etc
#4 factor out a sinx etc