trig question.?

Question

solve the equation.
sec^2x-secx-2=0

so HARD!

Answer 1

1 / cos ² x - 1 / cos x - 2 = 0
1 - cos x - 2 cos ² x = 0
2 cos ² x + cos x - 1 = 0
(2 cos x - 1) (cos x + 1) = 0
cos x = 1 / 2 , cos x = - 1
x = π / 3 , 5π / 3 , π

You can do it---don`t give in---good luck.

Answer 2

Quadratic (the unknown is secx)
u^2 - u - 2 = 0

sec x = [1 +/- SQRT(1+ 8) ] / 2

sec x = (1/2)* [ 1 +/- SQRT(9) ] = (1/2)*[1 +/- 3]

sec x = +2 and -1

which is the same as cos x = +0.5 and cos x = -1

x = 60 degrees (or pi/3) and
x = 180 degrees (or pi radians)

and all the cyclical equivalents

e.g.,
x = 2*k*pi +/- pi/3 (where k is any integer)

Answer 3

Hey there!

Use substitution. Let u=sec(x).

Here's the answer.

sec^2(x)-sec(x)-2=0 --> Write the problem.
u^2-u-2=0 --> Substitute u for sec(x).
(u-2)(u+1)=0 --> Completely factor u^2-u-2.
u-2=0 or u+1=0 --> Use the property, if pq=0, then p=0 or q=0.
u=2 or u=-1 --> Solve each equation for u.
sec(x)=2 or sec(x)=-1 --> Substitute sec(x) for u.
x=arcsec(2) or x=arcsec(-1) --> Take the inverse secant on both sides of the equation.
x=60, x=300 or x=180 Evaluate each equation.

So the answer is x=60, x=180, x=300. If you want to express it in radians, the answer is x=pi/3, x=pi and x=5pi/3.

Hope it helps!

Answer 4

Ok, let's use substitution to solve this:

Let's make sec(x) = u

So, we'll replace all of the sec(x)'s with u's:

u^2 - u - 2 = 0

Now it's just a quadratic equation, so it must be able to be factored:

(u + 1)(u - 2) = 0

The solutions of the equation are u = -1 and u = 2

So, let's replace the u's with sec(x)'s again:

sec(x) = -1 and sec(x) = 2

x = π and x = π/3, 5π/3

Answer 5

Try converting it into an algebraic expression.

Let sec x = y

Then sec² x = y², and you have

y² - y - 2 = 0

Solve:
(y - 2)(y + 1) = 0

y = 2 or -1

sec x = 2

1/cosx = 2

cos x = 1/2, Therefore, x = 60°.

or

sec x = -1

1/cos x = -1

cos x = -1

x = 180°

Answer 6

sec^2x -- secx -- 2 = 0
Or sec^2x -- 2secx + secx -- 2 = 0
Or secx(secx -- 2) + 1(secx -- 2) = 0
Or (secx -- 2)(secx + 1) = 0 giving
secx = 2, -- 1 = sec π/3, sec π
whence x = 2nπ ± π/3, 2nπ ± π

Answer 7

sec^2x-secx-2=0
Factor this into 2 backets, so we get
(secx-2)(secx+1)=0
Taking each of the 2 cases, we get

secx=2 or cosx= 1/2
x=pi/3, 5pi/3

secx=-1 or cosx= -1
x=pi

Answers for domain of 0

Answer 8

first, write in terms of cos, this is just easier for me...

1/cos^2x -1/cosx -2 =0

mult through by cos^2x:

1-cosx-2cos^2x=0 or
2cos^2x +cosx -1=0

treat this as a quadratic equation in cos x, and solve:


cosx=[-1+/-sqrt[1+8]]/2
cosx=-1, 1/2
x=180 degrees or 60 degrees (and also 300 degrees)

Answer 9

[sec(x) -2][sec(x) +1] = 0
sec x = 2 --> x = arcsec (2)
sec(x) = -1 --> x =arcsec(-1)