Finding all solutions to Trig Equations

• Apr 16th 2011, 09:15 AM
MajorJohnson
Finding all solutions to Trig Equations
I'm working on a few problems where I have to figure out all solutions to a given equation between [0,2PI). Here's one i got but i'm not all that sure that it is correct:

tan x ^ 2 - 2 tan x - 3 = 0

My Solution:

(tan x - 3) (tan x + 1) = 0

tan x = 3 , tan x = -1

Both have no solution.

I don't think this is right though, i've yet to solve any of these problems where i didn't find at least one solution. I'm guessing i would probably have to solve it another way correct?
• Apr 16th 2011, 09:46 AM
e^(i*pi)
3pi/4 is a solution to tan(x) = -1.
• Apr 16th 2011, 09:56 AM
Soroban
Hello, MajorJohnson!

Quote:

Find all solutions on [0, 2pi]

. . (tan x)^2 - 2(tan x) - 3 .= .0

My Solution:

. . (tan x - 3)(tan x + 1) .= .0

. . tan x =3 . . . tan x = -1

Both have no solution. . What?

. . x .= .arctan(3) . .0.40(pi)

. . x .= .arctan(-1) .= .0.75(pi)

• Apr 16th 2011, 02:10 PM
MajorJohnson
I don't seem to get those solutions when i enter in arctan.

I see where you get 3PI/4 from tangent and i think 7PI / 4 would be the next one.

As for 3 i'm not sure.
• Apr 16th 2011, 03:09 PM
topsquark
Quote:

Originally Posted by MajorJohnson
I don't seem to get those solutions when i enter in arctan.

I see where you get 3PI/4 from tangent and i think 7PI / 4 would be the next one.

As for 3 i'm not sure.

Two comments....first is your calculator in degree or radian mode? Also Soroban is giving his answers in terms of multiples of pi. To get his answer for atan(-1), for example, plug in atan(-1) to get -0.785398, then divide by pi = 3.1415927. You will get -0.25. I'll let you figure out how Soroban got 0.75. (Hint: Think about what quadrant the angle -1 rad is in.)

-Dan
• Apr 18th 2011, 05:07 AM
MajorJohnson
Question: Why did you divide by PI?

I was able to get that answer by doing what you described above.

Tan 3 is in quadrant 1 and 3.

so plugging in Tan (3) will give me -.14254... and i'll divide that by PI to give me -.04537...
• Apr 18th 2011, 05:15 AM
topsquark
Quote:

Originally Posted by MajorJohnson
I was able to get that answer by doing what you described above.

Next i have to find the decimal of each radian. For finding the decimal of a csc, sec, and cot i type on my calculator 1/sin(PI/6) right?

That's csc(pi/6), yes. then sec() = 1/cos() and cot() = 1/tan().

-Dan
• Apr 18th 2011, 05:56 AM
MajorJohnson

Why did you divide by PI?

Tan 3 would be in in quadrant 1 and 3.

so plugging in Tan (3) will give me -.14254... and i'll divide that by PI to give me -.04537...

But i still think that tan 3 would be present no solution because it is greater than 1. So the answer in radians would just be
3PI/4 and 7PI/4 right?
• Apr 18th 2011, 06:05 AM
mr fantastic
Quote:

Originally Posted by MajorJohnson

Why did you divide by PI?

Tan 3 would be in in quadrant 1 and 3.

so plugging in Tan (3) will give me -.14254... and i'll divide that by PI to give me -.04537...

But i still think that tan 3 would be present no solution because it is greater than 1. So the answer in radians would just be
3PI/4 and 7PI/4 right?

tan(x) can be any real number. You are confusing the range of tan with the range of sine and cosine.
• Apr 18th 2011, 06:17 AM
MajorJohnson
okay, so if 3 is acceptable, how would i find the radian measure on a 360 degree circle, i haven't learned how to solve it for this specific number only for ones up to 1.
• Apr 18th 2011, 01:43 PM
mr fantastic
Quote:

Originally Posted by MajorJohnson
okay, so if 3 is acceptable, how would i find the radian measure on a 360 degree circle, i haven't learned how to solve it for this specific number only for ones up to 1.

Posts # 2 and #3 gave you most of the answer. Below are all the solutions. You need to become more familiar with the symmetry of the unit circle, as well as the tan function. I suggest you thoroughly revise that material.

Quote:

Originally Posted by e^(i*pi)
3pi/4 is a solution to tan(x) = -1.

7 pi/4 is another.

And the solution to tan(x) = 3 is x = arctan(3) or x = pi + arctan(3).