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Find the number of solutions to $$\sec \theta + \csc \theta = \sqrt{15}$$ where $$0 \le \theta \le 2 \pi$$

Sep 27, 2019

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[ 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

And M = (-.412 + 4pi)rads                                    or     M  =  (.412 + 3 pi)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

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

Sep 28, 2019