Find an equation of the tangent line to the curve at the given point.

Question

y = sec (x) − 2 cos (x), P = (pi/3, 1)?

Answer 1

The curve is y =  sec (x) - 2 cos (x) and the point is (π/3, 1).

Differentiate the curve with respect to ' x '.

y ' = sec (x) tan (x) + 2sin (x)

When, x = π/3, y ' = sec (π/3) tan (π/3) + 2sin (π/3) = 2(√3) + 2(√3/2) = 2√3 + √3 = 3√3.

y ' = 3√3.

This is the slope (m ) of the tangent line to the implicit curve at (π/3, 1).

 

Slope - intercept form line equation is y = mx + b, where m is slope and b is y - intercept.

Slope (m) = 3√3.

Now, the tangent line equation is y = (3√3)x + b.

 

Find the y - intercept by substituting the the point in the tangent line equation say (x, y) = (π/3, 1).

1 = (3√3)(π/3) + b

⇒ b = 1 - √3π.

The tangent line equation  is y = (3√3)x + (1 - √3π).