Sin(45+x) = tan (45-x). Solve for x

Question

Grateful for any help

get to 

 

sqrt(2)/2(sin x + cos x) = (cos x - sin x) / (sin x + cos x). 

 

But it then dead end ??

Answer 1

This turned out to be pretty difficult, if you are in Algebra I suspect there may be a typo or this is intended to be really tough. OKAY!

Lets start by using the cofunction identity { cos(a)=sin(90-a) } to get the insides to match

I recommend always using cofunction identities when you see 90 degrees or 45 degrees in your problems

 

a = 45-x

sin (45+x) = cos ( 90 -(45+x) ) = cos (45-x)

 

Now we have:

 

cos(45-x) = tan(45-x)

 

Lets change into sines and cosines and simplify

 

cos(45-x) = sin(45-x)/cos(45-x)

 

cos²(45-x) = sin(45-x)

 

Theres an identity we know for cos² lets apply it and see what we have

 

1-sin²(45-x) = sin(45-x)

 

0 = sin²(45-x) + sin(45-x) -1

 

This looks like the quadratic 0 = u² + u - 1 (u = sin(45-x))

 

This is where the difficulty begins!

 

The solutions to that quadratic are [-1±sqrt(5)]/2

 

That means we have two options:

 

sin(45-x) =  [-1+sqrt(5)]/2 

 

sin(45-x) = [-1-sqrt(5)]/2   <- This one is impossible because [-1-sqrt(5)]/2 = -1.61 and sin is only between -1 and 1

 

SO! What we are left with is this:

 

sin(45-x) = [-1+sqrt(5)]/2

 

If we use arcsin as the inverse to reverse this equation we get:

 

45-x = arcsin ( [-1+sqrt(5)]/2 )

 

45-x = 38.17271

 

x = 6.82729°

 

I got to this point and assumed that I couldnt possibly be right but if you plug it into the original equation:

 

sin (45+x) = 0.7861514

tan(45-x) =  0.7861514

 

So there you have it

x = 6.82729°

 

Again if you are in Algebra, I would have to guess you misread the question or your teacher/book mistyped it. In any event, go ahead and turn this in if it made any sense to you, maybe you can get some extra credit. I could use some myself right about now. >_>