Answer to Question #1755 in Trigonometry for Isha

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'