Collaboration and help.

Question

Determine the exact value of cos(csc-1(-37-35))

Determine the exact value of Sin43 degrees cos77 degrees+cos 43degreess sin77degrees

Determine the exact value of cos165 degrees+cos105degrees

If sinA=5/13 and 90degrees< A <180degrees, and tanB= 5 and 180degrees <B <270degrees, determine the exact value of cos(B - A).

Answer 1

I don't follow the setup of the first problem with the -1 !!

The second problem is the trig identity for sin(a+b) so look at Patrick's answer.

The third problem is the trig identity for cosa+cosb.

cosa+cosb=2cos([a+b]/2)cos([a-b]/2)

You want the exact value.

2cos(270/2)cos(60/2)

2cos135cos30

look on the unit circle

2(-√2/2)(√3/2)

(-√2)(√3/2)

-√6/2 is the exact value

For the last one tanb=5, tanb=5/1 and tanb=opposite/adjacent

Therefore draw a right triangle and the opposite is 5, the adjacent is 1, and the hypotenuse is √26.

Now look at Patrick's work and the trig identity cos(b-a)=cosbcosa+sinbsina.

However, I got 37√26/(26*13)=37√26/338 for the exact value.

Hope I filled in some gaps for you.

Answer 2

cos(csc(72)) = cos(1/(sin 72))

sin(43+77) = sin(120) = sqrt(3)/2

cos 165 + cos 105 = -cos ( 15) - cos(75)

Finally sinA = 5/13 ---> cosA=12/13 and tanB = sin B/cosB = 5/1

which forces sin B = 5*sqrt(26)/26 and cos B = sqrt(26)/26

cos(B-a) = cosBcosA + sinB sin A

sqrt(26)/26 * (12/13) + 5 * sqrt(26)/26 * (5/13) =

17 * sqrt(26) / (26*13)