Give the general solution for the following trigonometric equation.

Question

-144 cos(y)-85= -1-252-sin(y)/tan(y)

#1 cos(y) =

"single trig ratio, equal to a constant (as a fraction)"

#2 r. a.=38.94

#3 Y=38.94+360k where k Є Z or y= -38.94+360 where k Є Z

I have some of the answers. I'm just confused about how to find the answer to question #1.

Answer 1

sin(y)/tan(y) = cos(y)

-144 cos(y)-85= -1-252sin(y)/tan(y)

-144cos(y) - 85 = -1 - 252cos(y)

108cos(y) = 84

cos(y) = 84/108 = 7/9

y = arccos(7/9) = about 38.94 degrees.

The domain of arccos is only the 1st and 2nd quadrants. If you want to extend this to [0,360) for the other value, it's 2pi - arccos(7/9) = about 5.60 radians = about 321.06 degrees.

If you want all values then it's about (in degrees) 38.94 + 360k and 321.06 + 360k