Help me! solve this equation: 8tan^2x -2secx-7=0 ?

Question

a) Find all solutions on intervals 0< x < 360
b) Find all degree solutions (round to two decimal places)

Answer 1

8 tan² x - 2 sec x - 7 = 0

First, multiply everything by cos² x:

8 sin² x - 2 cos x - 7 cos² x = 0

Using the Pythagorean theorem:

8 - 8 cos² x - 2 cos x - 7 cos² x = 0

Simplify:

-15 cos² x - 2 cos x + 8 = 0

Solve for cos x using the quadratic formula:

cos x = (2±√(4 + 4*15*8))/(-30) = (2±√484)/(-30) = -(2±22)/30
cos x = -24/30=-4/5 or cos x = 20/30 = 2/3

Now if cos x = -4/5 and 0° < x < 360°, then we have x=arccos (-4/5) or x=360° - arccos (-4/5). Similarly, if cos x = 2/3, the two solutions in (0°, 360°) are x=arccos (2/3) and x=360°-arccos (2/3). Plugging these into your calculator, you find numerical solutions of:

x=arccos (-4/5) ≈ 143.13°
x=360° - arccos (-4/5) ≈ 216.87
x=arccos (2/3) ≈ 48.19
x=360° - arccos (2/3) ≈ 311.81

And we are done.

Answer 2

Steps: a million) Distribute on the two aspects 2) combine like words on the two aspects 3) pass x variables to a minimum of one part, many times the left part 4) pass non-x variables to the different part, many times the magnificent part 5) divide the two aspects by the x's premier coefficient. 6) verify your artwork For the given project: 5(x-a million) = 3(x+4) a million) 5x - 5 = 3x + 12 2) Already achieved 3) 5x - 3x - 5 = 3x - 3x + 12 2x - 5 = 12 4) 2x - 5 + 5 = 12 + 5 2x = 17 5) 2x / 2 = 17 / 2 x = 17/2 6) Left part: 5(17/2 - a million) = 5(15/2) = seventy 5/2 ** 17/2 - a million = 17/2 - 2/2 = 15/2 precise part: 3 (17/2 + 4) = 3(25/2) = seventy 5/2 ** 17/2 + 4 = 17/2 + 8/2 = 25/2 the two aspects are equivalent, hence x = 17/2.

Answer 3

a)
8tan^2x - 2secx - 7 = 0
8[(secx)^2-1] - 2secx - 7 = 0
8(secx)^2 - 8 - 2secx - 7 = 0
8(secx)^2 - 2secx - 15 = 0
secx = [2 +- SQRT(4 - 4(8)(-15))] / 2(8)
secx = (2 +- 22)/16
secx = 24/16 or -20/16
1/cosx = 3/2 or -5/4
cosx = 2/3 or -4/5
x = 360n+-48.19, 360n+-143.13

b)
x = 48.19, 360-48.18; 143.13, 360-143.13
x = 48.19, 311.82; 143.13, 216.87

Answer 4

8tan^2x -2secx-7=0

8tan^2x -2*1/cosx-7=0
8sin²x/cos²x-2/cosx -7=0
8sin²x - 2cosx -7cos²x=0
8(1-cos²x)- 2cosx -7cos²x=0
-15cos²x- 2cosx +8=0
15cos²x+ 2cosx -8=0
(5cosx+4)(3cosx-2)=0

etc

Answer 5

8tan^2x -2secx-7=0
8(sin/cos)^2 - 2/cos - 7 = 0
left and right x cos^2
8sin^2 - 2cos - 7cos^2 = 0
8(1 - cos^2) - 2cos - 7cos^2 = 0
Now cos = y
8(1 - y2) - 2y - 7y2 = 0
-15y2 -2y - 7 = 0
15y2 + 2y + 7 = 0
15y2 + 2y + 7 = 0
Discr = 2 x 2 - 4 x 15 x 7 <0
No solution at all.

Th

Answer 6

8tan^2x -2secx-7=0
8sin^2 x/cos^2x - 2/ cosx - 7 = 0
8sin^2x -2cos x- 7cos^2x = 0
8(1-cos^2x) -2cosx -7cos^2x = 0
-15cos^2x -cosx +8 = 0
cos x = [1 +/- sqrt(1- 4(8)(-15)]/-30
cos x = -1/30 +/-sqrt(481)/-30
cos x = -.76439 and 0.69777237
x = arccos (-.76439) = +/-139.85 degrees
x =arccos(.69777) = +/- 45.75 degrees

Answer 7

You could just use the relations
tan^2 = sec^2 - 1
So 8*(sec^2 x - 1) - 2 secx - 7 = 0
Writing y = secx
8y^2 - 2y - 15 = 0
y = 1.5, -1.25

cosx = -0.8, 2/3

x = 143.13, 216.87, 48.19, 311.81 degrees

Answer 8

8 sin^2x/cos^2x-2/cosx-7=0
8sin^x-2cosx-7cos^2x=0
8(1-cos^2x)-2cosx-7cos^2x=0
8 -15cos^2x-2cosx=0
15cos^x+2cosx-8=0
(5cosx+4)(3cosx-2)=0
cosx=-4/5, cosx=2/3
rest is left for student.

Answer 9

8{tan(x)}^2 - 2sec(x) - 7 = 0
8[{sec(x)}^2 -1] - 2sec(x) - 7 = 0 [as {sec(x)}^2 - {tan(x)}^2 = 1]
8{sec(x)}^2 - 2sec(x) - 15 = 0
{2sec(x) - 3}{4sec(x) + 5} = 0