Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.)
cos(x+3pi/4) - cos(x-3pi/4) = 1
cos(x+3pi/4) = cosx cos(3π/4) - sinx sin(3π/4)
cos(x - (3π/4)) = cosx cos(3π/4) + sinx sin(3π/4)
cos(x+3pi/4) - cos(x-3pi/4) = 1
cosx cos(3π/4) - sinx sin(3π/4) - cosx cos(3π/4) - sinx sin(3π/4) = 1
-2sinx sin(3π/4) = 1
sinx (√2/2) = -1/2
sinx = (-1/2)(2/√2) = -1/√2
x = 225° or x = 315°
x = 5π/4 or x = 7π/4
right
To find the solutions of the equation cos(x+3pi/4) - cos(x-3pi/4) = 1 in the interval [0, 2π), we can follow these steps:
Step 1: Use the trigonometric identity cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2).
So, the equation can be rewritten as -2sin((x+3pi/4 + x-3pi/4)/2)sin((x+3pi/4 - x+3pi/4)/2) = 1.
Simplifying further, we get -2sin(x/(2pi))sin(3pi/4) = 1.
Step 2: Find the values of sin(3pi/4) and simplify the equation.
sin(3pi/4) = sqrt(2)/2.
So, the equation becomes -2sin(x/(2pi)) * (sqrt(2)/2) = 1.
Simplifying further, we have -sqrt(2)sin(x/(2pi)) = 1.
Step 3: Divide both sides of the equation by -sqrt(2) to isolate sin(x/(2pi)).
sin(x/(2pi)) = -1/sqrt(2).
Step 4: Find the values of x in the interval [0, 2π) for which sin(x/(2pi)) = -1/sqrt(2).
One solution for sin(x/(2pi)) = -1/sqrt(2) in the given interval is x/(2pi) = -pi/4.
Solving for x, we get x = -pi/2.
Step 5: To find the other solutions, we can use the periodicity of sine.
sin(x/(2pi)) has a period of 2pi.
Therefore, x/(2pi) = -pi/4 + 2nπ, where n is an integer.
Solving for x, we have x = -pi/2 + 4nπ, where n is an integer.
Step 6: Find all values of x in the interval [0, 2π).
In the given interval, [0, 2π), the values of n that satisfy this range are n = 0 and n = 1.
Substituting these values into x = -pi/2 + 4nπ, we get:
For n = 0: x = -pi/2 + 4(0)π = -pi/2.
For n = 1: x = -pi/2 + 4(1)π = 7pi/2.
Therefore, the solutions of the equation cos(x+3pi/4) - cos(x-3pi/4) = 1 in the interval [0, 2π) are x = -pi/2 and x = 7pi/2.
Final Answer: -pi/2, 7pi/2
To solve this equation, we will use trigonometric identities and properties. Let's break it down step by step:
1. Start by applying the identity: cos(a - b) = cos(a)cos(b) + sin(a)sin(b). This will help us simplify the equation.
So, the equation becomes:
cos(x+3pi/4) - cos(x-3pi/4) = 1
=> (cos(x)cos(3pi/4) + sin(x)sin(3pi/4)) - (cos(x)cos(-3pi/4) + sin(x)sin(-3pi/4)) = 1
2. Simplify further by using the values of cosine and sine of 3pi/4 and -3pi/4.
cos(3pi/4) = -sqrt(2)/2
sin(3pi/4) = sqrt(2)/2
cos(-3pi/4) = -sqrt(2)/2
sin(-3pi/4) = -sqrt(2)/2
Now, the equation becomes:
(cos(x)(-sqrt(2)/2) + sin(x)(sqrt(2)/2)) - (cos(x)(-sqrt(2)/2) + sin(x)(-sqrt(2)/2)) = 1
3. Simplify the equation further by combining like terms:
(-sqrt(2)/2)cos(x) + (sqrt(2)/2)sin(x) + (sqrt(2)/2)cos(x) - (sqrt(2)/2)sin(x) = 1
The terms with cos(x) and sin(x) cancel out, so we are left with:
-2sqrt(2)/2sin(x) = 1
4. Solve for sin(x):
We can rewrite the equation as:
-sqrt(2)sin(x) = 1 * 2sqrt(2)
This simplifies to:
sin(x) = -2
However, since the sine function has a range between -1 and 1, there are no solutions in the interval [0, 2π) for sin(x) = -2.
Therefore, the equation cos(x+3pi/4) - cos(x-3pi/4) = 1 has no solutions in the given interval.