Question

sin(x+45) = tanx, find x

Answer 1

Your username is supporting your question.. :P

Answer 2

Oh, from 18 minutes you have not got any help.. you wait has come to an end now.. :)

Answer 3

First of all:

\[\huge \color{green}{\textsf{Welcome To Openstudy...}}\]

Answer 4

\[\sin(A+B) = \sin(A) \cos(B) + \cos(A)\sin(B)\]

Answer 5

@ganeshie8

Answer 6

Umm..what??

Answer 7

\[\implies \sin(x) \cos(45) + \cos(x) \sin(45) = \tan(x) \\ \implies \frac{1}{\sqrt{2}}(\sin(x) + \cos(x)) = \tan(x) \\ \implies Doing \; \; wrong..\]

Answer 8

Nothing, just trying it.. :)

Answer 9

The fact that wolfram alpha doesn't provide a form with respect to x (and the solutions it does provide are approximations) would suggest that this is impossible:

http://www.wolframalpha.com/input/?i=sin(x%2B45)%3Dtan(x)

I am approaching it the same as @waterineyes and I think we are both stuck in the same place. I would be very interested in seeing a solution though, if it is possible.

Answer 10

Is the question from a textbook or did you make it up? :)

Answer 11

I should also change my username now.. :P

Answer 12

Oh its from

some question paper
There's no answer key...
The question might be wrong...

Answer 13

what approximation methods are you allowed to use ?

Answer 14

I'm not sure if other people noticed this but, by inspection, one solution is 45 degrees; this guy gives an awesome solution.

https://answers.yahoo.com/question/index?qid=20130304110356AAQtCWL

Answer 15

nice :)

Answer 16

x must be between 0 to 45 degrees.

Answer 17

Ik x is 45 but idk how to get x as 45..

Answer 18

Do you understand the solution given at Yahoo Answers or would you like an extra explanation?

https://answers.yahoo.com/question/index?qid=20130304110356AAQtCWL

Answer 19

And we aren't allowed to use any approximation methods..I'm still learning basic trig..

Answer 20

Haha, 'basic' trig. Have you had a look at the Yahoo answers solution? Does it make sense?

I would say that if you are learning "basic" trig then they would only be expecting you to look at the function and realise that 45 degrees would work.

Answer 21

It kind of does, kind of doesn't....
But we can't just write 45..I mean there must be some way to solve it...

Answer 22

There is but it is pretty difficult, the Yahoo answers solution is the only one that I can see.

What grade are you in, in what country? (I am wondering what background knowledge you have)

Answer 23

Well thanks anyways and I'm still in tenth so...
I guess I'll just write 45.

Answer 24

Nice work by Brian there.. :)

Answer 25

Just by inspection we can see that if x = 45, then sin(45 + 45) = sin(90) = 1
and tan(45) = 1, so x = 45 degrees is one solution.
(and in general (45 + n*360) degrees). But we should look to see if there are others.
sin(x + 45) = sin(x)*cos(45) + cos(x)*sin(45) = (1/sqrt(2))*(sin(x) + cos(x)).
So for this to equal tan(x) we require that
sin(x) + cos(x) = sqrt(2)*tan(x) ---->
sin(x) + cos(x) = sqrt(2)*sin(x)/cos(x) ---->
sin(x)*cos(x) + cos^2(x) = sqrt(2)*sin(x) ---->
cos^2(x) = (sqrt(2) - cos(x))*sin(x) ----> square both sides to get
cos^4(x) = (2 - 2*sqrt(2)*cos(x) + cos^2(x))*sin^2(x) ----->
cos^4(x) = (2 - 2*sqrt(2)*cos(x) + cos^2(x))*(1 - cos^2(x)) ---->
cos^4(x) = 2 - 2*sqrt(2)*cos(x) - cos^2(x) + 2*sqrt(2)*cos^3(x) - cos^4(x) ---->
2*cos^4(x) - 2*sqrt(2)*cos^3(x) + cos^2(x) + 2*sqrt(2)*cos(x) - 2 = 0.
Letting z = cos(x) the polynomial can be written as
2*z^4 - 2*sqrt(2)*z^3 + z^2 + 2*sqrt(2)*z - 2 = 0.
We need numerical techniques to solve this.Using Newton's Method I get
z = cos(x) = 0.707, (which gives the answer I guessed at initially),
and z = cos(x) = -0.930, which gives us x = +/- 158.4 degrees + n*360 degrees.
Testing this in the original equation, we find that only x = (158.4 + n*360) degrees
works. So on [0, 360 degrees] the solutions are {45, 158.4 degrees}.
(Note that the 158.4 degrees is only accurate to 1 decimal place.)