sin(x+(pi/4))- sin(x-(pi/4))=1
1. Apply the sum and difference formulas for sine to simplify the left-hand side of the equation.
2. find all the solutions you found to the equation you found.
Please help me. Thank you.
To solve the equation sin(x+(pi/4))-sin(x-(pi/4)) = 1, we can follow these steps:
1. Apply the sum and difference formulas for sine:
Using the sum formula for sine (sin(A+B) = sin(A)cos(B) + cos(A)sin(B)) and the difference formula for sine (sin(A-B) = sin(A)cos(B) - cos(A)sin(B)), we have:
sin(x+(pi/4)) - sin(x-(pi/4))
= (sin(x)cos(pi/4) + cos(x)sin(pi/4)) - (sin(x)cos(pi/4) - cos(x)sin(pi/4))
= 2cos(x)sin(pi/4)
= sqrt(2)*cos(x) [because sin(pi/4) = cos(pi/4) = sqrt(2)/2]
So, the equation becomes: sqrt(2)*cos(x) = 1.
2. Find all the solutions:
To find the values of x that satisfy the equation sqrt(2)*cos(x) = 1, we can isolate cos(x) and then solve for x.
Divide both sides of the equation by sqrt(2):
cos(x) = 1/sqrt(2) = sqrt(2)/2.
Now, we need to find the angles whose cosine is sqrt(2)/2. These angles lie in the first and fourth quadrants, where the cosine function is positive.
In the first quadrant (0 to pi/2), the angle x that satisfies cos(x) = sqrt(2)/2 is x = pi/4.
In the fourth quadrant (2pi to 5pi/2), the angle x that satisfies cos(x) = sqrt(2)/2 is x = 7pi/4.
Hence, the solutions to the equation sin(x+(pi/4))-sin(x-(pi/4)) = 1 are x = pi/4 and x = 7pi/4.
To simplify the left-hand side of the equation sin(x+(π/4)) - sin(x-(π/4)) = 1, we can apply the sum and difference formulas for sine.
The sum formula for sine states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B), and the difference formula for sine states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
Using these formulas, let's simplify the left-hand side of the equation step by step:
sin(x + (π/4)) - sin(x - (π/4))
= sin(x)cos(π/4) + cos(x)sin(π/4) - sin(x)cos(π/4) + cos(x)sin(π/4)
= cos(π/4)(sin(x) + cos(x)) + sin(π/4)(cos(x) - sin(x))
Now that we have simplified the left-hand side of the equation, our equation becomes:
cos(π/4)(sin(x) + cos(x)) + sin(π/4)(cos(x) - sin(x)) = 1
To find all the solutions to this equation, we can set each term inside the parentheses equal to a variable. Let's say sin(x) + cos(x) = a and cos(x) - sin(x) = b.
Now we can express the equation in terms of a and b:
cos(π/4)a + sin(π/4)b = 1
At this point, the equation can be represented as a linear equation in the variables a and b. We can solve it using algebraic methods or graphical methods to find the values of a and b.
Once we have found the values of a and b, we can substitute them back into the equations sin(x) + cos(x) = a and cos(x) - sin(x) = b to find the corresponding values of x. These values will be the solutions to the original equation sin(x + (π/4)) - sin(x - (π/4)) = 1.