To find the equation of the tangent line to the curve y = sin(x) at a given point, we will perform the following steps:
- Determine the point of tangency: In this case, the point given is (p, 0), where we need to ensure that y = sin(p) should equal 0. For this to be true, p must be one of the values where sine function equals zero, which occurs at p = nπ, where n is an integer.
- Find the derivative of the function: The derivative of y = sin(x) is y’ = cos(x). This derivative will give us the slope of the tangent line at the given point.
- Evaluate the derivative at point p: Substitute p into the derivative to find the slope of the tangent line:
m = cos(p)
- Use the point-slope form of the line: The point-slope form of a line is given by the formula y – y_1 = m(x – x_1), where (x_1, y_1) is the point of tangency, here (p, 0). Substituting the values we have, we get:
- Equation of the tangent line:
y – 0 = cos(p)(x – p)
Which simplifies to:
y = cos(p)(x – p)
Therefore, the equation of the tangent line to the curve y = sin(x) at the point (p, 0) is:
y = cos(p)(x – p)