Question

Question #78f22

Answer 1

`4^@09; 122^@77`

Explanation:

`cos x + 2sin x = pi/3`
Call `tan t = (sin t)/(cos t) = 2` --> `t = 63^@43`--> `cos t = 0.45`
The equation becomes:
`cos x + (sin t)/(cos t)sin x = pi/3`
`cos x.cos t + sin t.sin x = (pi/3)cos t = (3.41)(0.45)/3 = 0.51`
Using trig identity: cos (a - b) = cos a.cos b + sin a.sin b, we get:
cos (x - t) = cos (x - 63.43) = 0.51
Calculator and unit circle give:
`x - 63.43 = +- 59^@34` --> 2 solutions --
`x = 63.43 + 59.34 = 122^@77`
`x = 63.43 - 59.34 = 4^@09`
Check by calculator:
x = 122.77 --> cos x = -0.54 --> 2sin x = 1.68
`cos x + 2sin x = -0.54 + 1.68 = 1.141 = pi/3 = 3.414/3.` OK
x = 4.09 --> cos x = 0.99 --> 2sin x = 0.14
`cos x = 2sin x = 0.99 + 0.14 = 1.14 = pi/3.` OK