cos(3π/4+x) + sin (3π/4 -x) = 0
= cos(3π/4)cosx + sin(3π/4)sinx + sin(3π/4)cosx - cos(3π/4)sinx
= -1/sqrt2cosx + 1/sqrt2sinx + 1/sqrt2cosx - (-1/sqrt2sinx)
I canceled out -1/sqrt2cosx and 1/sqrt2cosx
Now I have
1/sqrt sinx + 1/sqrt2 sinx
And that doesn't equal 0. So where did I go wrong?
Also cos(x+y)cosy + sin(x+y)siny = cosx
I ended up with
(cosxcosy) + sinxsinycosy + (sinxcosy) + cosxsin^2y
I don't know what to do next.
Try your sum and difference identities again. I don't believe you've expanded them correctly.
cos(x + y) = cosx * cosy - sinx * siny
cos(x - y) = cosx * cosy + sinx * siny
sin(x + y) = sinx * cosy + cosx * siny
sin(x - y) = sinx * cosy - cosx * siny
cos(3π/4+x)
To solve the equation cos(3π/4+x) + sin (3π/4 -x) = 0, let's go step by step.
Starting from where you left off:
= -1/sqrt2cosx + 1/sqrt2sinx + 1/sqrt2cosx - (-1/sqrt2sinx)
When adding the last two terms, you made a mistake in the sign. It should be:
= -1/sqrt2cosx + 1/sqrt2sinx + 1/sqrt2cosx + 1/sqrt2sinx
= (-1/sqrt2cosx + 1/sqrt2cosx) + (1/sqrt2sinx + 1/sqrt2sinx)
= 0 + 2/sqrt2sinx
= 2/sqrt2sinx
Now, we have 2/sqrt2sinx, which is not equal to 0. Therefore, the equation cos(3π/4+x) + sin (3π/4 -x) = 0 does not have a solution.
Moving on to the other equation cos(x+y)cosy + sin(x+y)siny = cosx:
(cosxcosy) + sinxsinycosy + (sinxcosy) + cosxsin^2y
To simplify further, we can group the terms with cosine and sine as follows:
= (cosxcosy + sinxcosy) + (sinxsinycosy + cosxsin^2y)
= cos(x+y)cosy + sin(x+y)cosy + sinxsinycosy + cosxsin^2y
Now, let's focus on the terms with cosine:
= cos(x+y)cosy + sin(x+y)cosy
Using the identity cos(x+y) = cosxcosy - sinxsiny, we can rewrite the equation as:
= (cosxcosy - sinxsiny)cosy + sin(x+y)cosy
= cosxcos^2y - sinxsinycosy + sin(x+y)cosy
This is the simplified form of the equation. There is no need to perform any further simplifications.
I hope this helps clarify your questions! Let me know if you have any further doubts.
In your first equation, you correctly applied the sum of angles formula to expand cos(3π/4 + x) and sin(3π/4 - x). However, there seems to be a mistake in your calculation.
Let's go over the expansion step by step:
cos(3π/4 + x) + sin(3π/4 - x) = (cos(3π/4)cos(x) - sin(3π/4)sin(x)) + (sin(3π/4)cos(x) - cos(3π/4)sin(x))
Now let's simplify each term separately:
= - (1/√2)cos(x) - (1/√2)sin(x) + (1/√2)cos(x) - (1/√2)sin(x)
= -(2/√2)sin(x)
= -√2sin(x)
So the correct simplified expression is -√2sin(x), not 1/√2sin(x).
Regarding your second equation, we have:
cos(x+y)cos(y) + sin(x+y)sin(y) = cos(x)
To simplify this expression, we can again apply the sum of angles formula to the left side:
(cos(x)cos(y) - sin(x)sin(y))cos(y) + (sin(x)cos(y) + cos(x)sin(y))sin(y) = cos(x)
Expanding further:
cos(x)cos^2(y) - sin(x)sin(y)cos(y) + sin(x)cos(y)sin(y) + cos(x)sin^2(y) = cos(x)
Now, let's group the similar terms together:
cos(x)(cos^2(y) + sin^2(y)) + sin(x)(sin(y)cos(y) + sin(y)cos(y)) = cos(x)
Using the trigonometric identity cos^2(y) + sin^2(y) = 1:
cos(x)(1) + sin(x)(sin(y)cos(y) + sin(y)cos(y)) = cos(x)
Simplifying further:
cos(x) + 2sin(x)sin(y)cos(y) = cos(x)
At this point, we have obtained an equation involving cos(x), sin(x), sin(y), and cos(y). And if you want to further simplify it or solve for a specific variable, you may need additional information or the context in which this equation arises.