Question

Solve this trigonometric equation.give all positive values of the angle between 0deg and 360deg that will satisfy.
Give any approximate value to the nearest minute only.


& tan (x+15) = 3 tanx

Answer 1

tan(x+15) = (tanx + tan15)/(1-tanx tan15)
(tanx + tan15)/(1-tanx tan15) = 3 tanx
tanx + tan 15 = 3 tanx - 3tan2x tan15
3 tan²x tan15 - 2tanx + tan15 = 0
tan15 = 0.27, denote tanx as t
0.81 t² - 2t + 0.27 = 0
D = 0.1252
t₁ = 1.67
t₂ = 0.80
tan (x₁) = 1.67
x₁ = arctan (1.67) + πn = 59.087 + 180n = 59º 5'+ 180n, n=0,1,2,..
tan(x₂) = 0.80
x₂ = arctan(0.80) + πn = 38.66 + 180n = 38º 40' + 180n, n=0,1,2,..

Answer: 59º 5', 239º 5', 38º 40', 218º 40'