How do I find the equation of a tangent of y=cos(2x+(pi)/3) at x=(pi)/4
To find the equation of the tangent to a curve at a specific point, you need to determine the slope of the tangent at that point and the coordinates of the point. Here are the steps to find the equation of the tangent to the curve y = cos(2x + π/3) at x = π/4:
Step 1: Take the derivative of the given function to find the slope of the tangent line. The derivative of cos(u) is -sin(u), and using the chain rule, we differentiate 2x + π/3 as 2.
dy/dx = -sin(2x + π/3) * 2
Step 2: Plug x = π/4 into the derivative to find the slope of the tangent line at that point.
dy/dx = -sin(2 * (π/4) + π/3) * 2
= -sin(π/2 + π/3) * 2
= -sin(5π/6) * 2
= -√3/2 * 2
= -√3
So, the slope of the tangent line is -√3.
Step 3: Plug the given x-value (π/4) into the original function to find the corresponding y-coordinate.
y = cos(2 * (π/4) + π/3)
= cos(π/2 + π/3)
= cos(5π/6)
= -1/2
Thus, the point of tangency is (π/4, -1/2).
Step 4: Now that you have the slope (-√3) and the coordinates of the point (π/4, -1/2), you can use the point-slope form to find the equation of the tangent line.
The point-slope form is: y - y1 = m(x - x1)
Substituting the values, we get:
y - (-1/2) = -√3(x - π/4)
Simplifying the equation:
y + 1/2 = -√3x + √3(π/4)
y + 1/2 = -√3x + √3π/4
Therefore, the equation of the tangent to the curve y = cos(2x + π/3) at x = π/4 is:
y = -√3x + √3π/4 - 1/2