Solve for:
1+cosx/sinx + sinx/1+cosx = 1
My teacher said no solution, but I don't know how he got that.
(1+cosx)/sinx + sinx/(1+cosx)
=
(1+cosx)^2 + sin^2x
------------------------
sinx(1+cosx)
1+2cosx+cos^2x+sin^2x
-------------------------
sinx(1+cosx)
= (2+2cosx)/(sinx(1+cosx))
= 2/sinx
so, does
2/sinx = 1
have any solutions?
cos x/sin x + sin x/(1+cos x) = 0 ??
nah, maybe you mean
(1+cos x)/sin x + sin x/(1+cos x)=1
(1+cos)(1-cos)/(sin(1-cos))
+sin (1-cos)/[(1+cos)(1-cos)] = 1
(1-cos^2)/(sin-sin cos)
+ (sin-sin cos)/(1-cos^2) = 1
sin/(1-cos) + (1-cos)/sin = 1
sin^2 + (1-cos)^2 = sin(1-cos)
(1-cos^2) +1 - 2 cos + cos^2 = sin-sin cos
2 - 2 cos = sin (1-cos)
humm
2 ( 1-cos) = sin(1-cos)
sin x = 2 !!!!!
well
-1 </= sin x </= +1
so impossible
Do I get extra credit for finding a harder way ?
To solve the equation (1 + cosx)/(sinx) + (sinx)/(1 + cosx) = 1, we can begin by simplifying the expression.
Let's denote A = (1 + cosx)/(sinx), and B = (sinx)/(1 + cosx).
So, the equation can be written as A + B = 1.
To eliminate fractions, we can multiply everything by sinx(1 + cosx). We get:
(sin^2)x + (sinx)(cosx) + sinx = sinx(1 + cosx).
Now, let's rearrange the equation:
(sin^2)x + (sinx)(cosx) + sinx - sinx(1 + cosx) = 0.
Simplifying further, we have:
(sin^2)x - sinx(cosx) - sinx(cosx) = 0.
Factoring out sinx, we get:
sinx(sin(x) - cosx - cosx) = 0.
This equation can be true in one of two cases:
1. sinx = 0.
If sinx = 0, then x = nπ, where n is an integer.
2. sin(x) - cosx - cosx = 0.
Rearranging, we have sin(x) = 2cosx, which can be rewritten as tan(x) = 2.
To find the values of x, we need to consider the solutions in the range of one period of the tangent function. So, we find the values of x in the interval [0, 2π).
Using a calculator or other mathematical tools, we find that there are two solutions in this interval:
- x ≈ 15.96 degrees (or x ≈ 0.28 radians)
- x ≈ 195.96 degrees (or x ≈ 3.43 radians)
Combining the solutions from both cases, we find that the possible values of x are:
x = nπ, where n is an integer, or
x ≈ 0.28 radians or x ≈ 3.43 radians.
So, the equation has solutions.
If your teacher said that there is no solution, it is possible that there was a mistake made during the calculation or interpretation of the answer. It is always a good idea to verify the solution independently or ask your teacher for clarification.