Solve: sin2xsinx - cos2xcosx = 1, x domain [-pi, pi].
or
cos2xsinx - sin2xcosx = -1
compare this to
cos(A+B) = cosAcosB - sinAsinB
looks like somebody expanded cos(2x+x)
so cos 3x = -1 , (I know cos π = -1)
3x = π
x = π/3
so for your given domain, x = ± π/3
To solve the equation sin(2x)sin(x) - cos(2x)cos(x) = 1, we will use the trigonometric identity:
sin(A - B) = sinAcosB - cosAsinB.
Let's rewrite the equation using this identity:
sin(2x - x) = 1
Simplifying further,
sin(x) = 1.
To find the solutions for x in the given domain of [-π, π], we can use the inverse sine function (sin^(-1)).
Taking the inverse sine of both sides, we get:
x = sin^(-1)(1)
Since the sine function ranges between -1 and 1, the only value of x that satisfies this equation is when sin(x) = 1.
Therefore, the solution is x = π/2.
Thus, the solution for the equation sin(2x)sin(x) - cos(2x)cos(x) = 1, in the domain [-π, π], is x = π/2.
To solve the equation sin(2x)sin(x) - cos(2x)cos(x) = 1 for the given domain [-π, π], we can use trigonometric identities to simplify the equation. One useful identity is the product-to-sum identity:
sin(A)sin(B) = 1/2 [cos(A-B) - cos(A+B)]
So, let's rewrite the given equation using this identity:
1/2 [cos(2x-x) - cos(2x+x)] - cos(2x)cos(x) = 1
Simplifying this further, we get:
1/2 [cos(x) - cos(3x)] - cos(2x)cos(x) = 1
Combine the terms that involve the same trigonometric function:
1/2 cos(x) - 1/2 cos(3x) - cos(2x)cos(x) = 1
Now, let's apply another trigonometric identity, the double angle formula:
cos(2θ) = 2cos²(θ) - 1
Using this identity, we can rewrite cos(2x)cos(x) as:
cos(2x)cos(x) = (2cos²(x) - 1)cos(x)
Substituting this back into the equation, we have:
1/2 cos(x) - 1/2 cos(3x) - (2cos²(x) - 1)cos(x) = 1
Now, let's simplify the equation further:
1/2 cos(x) - 1/2 cos(3x) - 2cos³(x) + cos(x) = 1
Combining like terms:
1.5 cos(x) - 1/2 cos(3x) - 2cos³(x) = 1
Rearranging the terms:
2cos³(x) - 1.5 cos(x) + 1/2 cos(3x) = -1
We now have an equation that involves multiple cosine terms. To solve this equation, we can use numerical methods such as graphing or iteration techniques.