4sin x = 3tan x
Call tan x/2 = t. Use the trig identities in terms of tan x/2.
sin x = (2t)/(1 + t^2) and tan x = (2t)/(1 - t^2)sinx=2t1+t2andtanx=2t1−t2
8t(1 - t^2) = 6t(1 + t^2)8t(1−t2)=6t(1+t2)
8t - 8t^3 = 6t + 6t^38t−8t3=6t+6t3
f(t) = 14t^3 - 2t = 2t(7t^2 - 1) = 2t(sqr7.t - -1)(sqr7.t + 1)f(t)=14t3−2t=2t(7t2−1)=2t(sqr7.t−−1)(sqr7.t+1)
f(x) = 4sin x - 3tan x = 0 when:
a. t = tan x/2 = 0 -> x/2 = pi -> x = 2pi
b. #t = tan x/2 = 1/sqr7 = 0.38 = tan 20.70 -> x = 41.40 deg
c. #t = tan x/2 = -1/sqr7 = -0.38 = tan (-20.70) -> x = -41.40 deg
Check:
x = 2pi -> sin x = 0; tan x = 0 x=2π→sinx=0;tanx=0 Correct
x = 41.40 -> sin x = 0.66 -> 4sin x = 2.64 x=41.40→sinx=0.66→4sinx=2.64 -> 3tan x = 2.64.3tanx=2.64. OK
x = -41.40 -> sin x = -0.66 --> 4sin x = -2.64 x=−41.40→sinx=−0.66−→4sinx=−2.64 -> 3tan x = -2.64 3tanx=−2.64. OK
