## if 3 sec²θ-5tan θ-4=0 find the general solution to this equation

Question

if 3 sec²θ-5tan θ-4=0 find the general solution to this equation

in progress
0

Mathematics
7 days
2021-07-16T07:20:09+00:00
2021-07-16T07:20:09+00:00 1 Answers
7 views
0
## Answers ( )

3 sec²(

θ) – 5 tan(θ) – 4 = 0Recall the Pythagorean identity,

cos²(

θ) + sin²(θ) = 1.Multiplying both sides by 1/cos²(

θ) gives another form of the identity,1 + tan²(

θ) = sec²(θ).Then the equation becomes quadratic in tan(

θ):3 (1 + tan²(

θ)) – 5 tan(θ) – 4 = 03 tan²(

θ) – 5 tan(θ) – 1 = 0I’ll solve by completing the square.

tan²(

θ) – 5/3 tan(θ)) – 1/3 = 0tan²(

θ) – 5/3 tan(θ) = 1/3tan²(

θ) – 5/3 tan(θ) + 25/36 = 1/3 + 25/36(tan(

θ) – 5/6)² = 37/36tan(

θ) – 5/6 = ±√37/6tan(

θ) = (5 ± √37)/6Take the inverse tangent of both sides:

θ= arctan((5 + √37)/6) +nπorθ= arctan((5 – √37)/6) +nπwhere

nis any integer