Find the number of solutions to \(\sec \theta + \csc \theta = \sqrt{15}\) where \(0 \le \theta \le 2 \pi\)
[ I'm using "A" for theta ]
sec A + csc A = sqrt (15)
1/cosA + 1/sinA = sqrt (15)
[sin A + cosA ]
___________ = sqrt (15)
sin A cos A
sinA + cosA = sqrt (15) * sinAcosA square both sides
sIn^2A + 2sinAcosA + cos^2A = 15sin^2Acos^2A
1 + 2sinAcosA = 15sin^2Acos^2A rearrange as
15sin^2Acos^2A - 2sinAcosA - 1 = 0 factor as
(5sinAcosA + 1) (3sinAcosA - 1) = 0
We have either that
5sinAcosA + 1 = 0
5sinAcosA = -1
sinAcosA = -1/5
(1/2)sin(2A) = -1/5
sin(2A) = -2/5
Let 2A = M
sin (M) = -2/5
Using the sine inverse
arcsin (-2/5) = M ≈ ( - .412 + 2pi)rads or M = (.412 + pi) rads
So 2A =( -.412 + 2pi) rads 2A = (.412 + pi)rads
A = -.206 + pi rads ≈ 2.94 rads A = .206 + pi/2 rads ≈ 1.77 rads
And M = (-.412 + 4pi)rads or M = (.412 + 3 pi)rads
2A = (-.412 + 4pi)rads 2A = (.412 + 3pi) rads
A = -..206 + 2pi rads ≈ 6.07 rads A = .206 + (3/2)pi rads ≈ 4.91 rads
Or
3sinAcosA - 1 = 0
3sinAcosA = 1
sinAcosA = 1/3
(1/2)sin (2A)= 1/3
sin(2A) = 2/3
Let 2a = M
sin(M) = 2/3
arcsin(2/3) = M ≈ .73 rads
So
2A = .73 rads or 2A = .(pi -.73) rads
A ≈ .365 rads A ≈( pi/2 - .365)rads ≈ 1.205 rads
Because of squaring....some of these solutions are extraneous.....as this graph shows : https://www.desmos.com/calculator/ucfunqh12v
The correct solutions from [0, 2pi] are
A ≈ [ .365 , 1.206, 2.94, 4.91 ] rads