find the values of x between 0 and 2 pi where the tangent line to the graph of y = sinxcosx is horizontal
Hmmmm. Wondering if you are taking calculus.
Calculus
y=sinxcosx
y'=0=cos^2x-sin^2x
cos^2x=sin^2x
tan^2 x=1
tanx= +-1
which occurs at increments of 45 degrees
x= +-PI/4, PI+-Pi/4
check that
To find the values of x between 0 and 2π where the tangent line to the graph of y = sin(x)cos(x) is horizontal, we need to find the points on the graph where the derivative of y with respect to x equals 0.
Let's start by finding the derivative of y = sin(x)cos(x) using the product rule:
dy/dx = cos(x)cos(x) - sin(x)(-sin(x))
= cos^2(x) + sin^2(x)
= 1
Now, we need to set the derivative equal to 0 and solve for x:
1 = 0
Since this equation has no solutions, there are no points on the graph where the tangent line is horizontal between 0 and 2π.
Hence, there are no values of x between 0 and 2π where the tangent line to the graph of y = sin(x)cos(x) is horizontal.
To find the values of x between 0 and 2π where the tangent line to the graph of y = sin(x)cos(x) is horizontal, we need to find the critical points where the derivative of the function is equal to zero.
To do this, we first differentiate the given function with respect to x:
dy/dx = (cos(x)cos(x)) + (sin(x)(-sin(x))) [using the product rule]
Now simplify the equation:
dy/dx = cos^2(x) - sin^2(x)
To find the critical points, set the derivative equal to zero and solve for x:
cos^2(x) - sin^2(x) = 0
Using the trigonometric identity cos^2(x) - sin^2(x) = cos(2x), we can rewrite the equation as:
cos(2x) = 0
Now we need to find the values of x between 0 and 2π where cosine is equal to zero. Since cosine is equal to zero at π/2 and 3π/2, we have two critical points:
2x = π/2 and 2x = 3π/2
Solving for x in each of these equations:
For 2x = π/2, x = π/4
For 2x = 3π/2, x = 3π/4
Since we are looking for values of x between 0 and 2π, we need to check if these critical points fall within this interval. Both π/4 and 3π/4 meet this requirement.
Therefore, the values of x between 0 and 2π where the tangent line to the graph of y = sin(x)cos(x) is horizontal are x = π/4 and x = 3π/4.