Question

Find the general solutions (a)3 cos x - 4 sin x = 2 (b) 3 cos x + 4 sin x =5 (c) 5 sin x - 12 cos x =13 (d) 2 sin 2x + cos 2x = 2?

Answer 1

Solve `3cos x - 4sin x = 2`

Ans: 13.30 and -119.55 deg

Explanation:

Divide both sides by 3:
`cos x - (4/3)sin x = 2/3`.
Call `tan a = sin a/cos a = 4/3` --> `a = 53.13` deg
`cos x.cos a - sin a.sin x = (2cos a)/3 = 0.6(2/3) = 0.4`
cos (x + a) = cos (x + 53.13) = 0.4 --> `+- 66.42` deg

a. x + 53/13 = 66.42 --> x = 66.42 - 53.13 = `13.30` deg
b. x + 53.13 = - 66.42 --> x = -66.42 - 53.13 = `- 119.55` deg
Check by calculator.
x = 13.30; 3cos x = 3(0.97) = 2.92; 4sin x = 4(0.23) = 0.92.
Therefor" 2.92 - 0.92 = 2. OK
x = - 119.55; 3cos x = 3(-0.49) = -1.48; 4sin x = -0.87(3) = -3.48.
Therefor, -1.48 - (- 3.48) = 2. OK

Answer 2

Solve: 2sin 2x + cos 2x = 2.

Ans: 18.15 and 45.28 deg

Explanation:

Divide both sides by 2:
`sin 2x + (1/2)cos 2x = 1`. Call tan a = sin a/cos a = 1/2# --> --> a = 26.57 deg. sin 2x.cos a + sin a.cos 2x = cos a = 0.89 sin (2x + 26.57) = 0.89 --> sin 62.87 and sin 117.13 a. 2x + 26.57 = 62.87 --> 2x = - 26.57 = 36.30 deg -> --> #x = 18.15# deg b. 2x + 26.57 = 117.13 --> 2x = 117.13 - 26.57 = 90.56 deg--> --> #x = 45.28# deg