2) Use the sum and difference identites
sin[x + pi/4] + sin[x-pi/4] = -1
sinx cospi/4 + cosxsin pi/4 + sinx cos pi/4 - cosx sin pi/4 = -1
2 sin x cos pi/4 =-1
cos pi/4 = sqr2/2
2sin^x(sqrt2/2) = -1
sin x = -sqrt2
x = 7pi/4 and 5pi/4
Am I correct?
2sin^x(sqrt2/2) = -1
sin x = -1/sqrt2 = - sqrt 2 /2
x = pi + pi/4 = 5 pi/4
x = 2 pi - pi/4 = 7 pi /4
so I agree with you but you seem to have a typo
To confirm if you are correct, let's analyze step by step the solution process.
Starting from the given equation:
sin[x + pi/4] + sin[x - pi/4] = -1
We can use the sum and difference identities for sine to simplify this expression:
sin(x)cos(pi/4) + cos(x)sin(pi/4) + sin(x)cos(pi/4) - cos(x)sin(pi/4) = -1
Simplifying further:
2sin(x)cos(pi/4) = -1
Since cos(pi/4) = sqrt(2)/2, we substitute this value:
2sin(x)(sqrt(2)/2) = -1
sin(x) = -1/(2 * sqrt(2)/2)
sin(x) = -1/sqrt(2)
sin(x) = -sqrt(2)/2
At this point, you wrote "sin x = -sqrt2". However, the correct value for sin(x) is -sqrt(2)/2.
Now, to find the values of x that satisfy this equation, we look for angles whose sine is -sqrt(2)/2. These angles can be found in the third and fourth quadrants.
In the third quadrant (where sin is negative), the angle x is:
x = 7pi/4
In the fourth quadrant (also where sin is negative), the angle x is:
x = 5pi/4
Therefore, the correct solutions for the equation are x = 7pi/4 and x = 5pi/4.
In conclusion, you are correct in your solution.