How do you solve secxtanx=2/3?

1 Answer
Nghi N
Nov 12, 2016

pi/6, (5pi)/6

Explanation:

sec xtan x = 2/3
(1/cos x)(sin x/(cos x)) = 2/3
Cross multiply:
3sin x = 2 cos^2 x = 2(1 - sin^2 x) = 2 - 2sin^2 x
Solve this quadratic equation in sin x:
2sin^2 x + 3sin x - 2 = 0.
Use the improved quadratic formula (Socratic Search)
D = d^2 = b^2 - 4ac = 9 + 16 = 25 --> d = +- 5
There are 2 real roots:
sin x = -b/(2a) +- d/(2a) = -3/4 +- 5/4
sin x = - 8/4 = - 2 (rejected because < - 1) and
sin x = 2/4 = 1/2
Trig table and unit circle -->
sin x = 1/2 --> arc x = pi/6 and arc x = (5pi)/6
For general answers, add (k2pi)

Related questions
See all questions in Solving Trigonometric Equations
Impact of this question
4181 views around the world
You can reuse this answer
Creative Commons License