Question

Find the equation of the tangent line to y=sinx at x=2pie

Answer 1

y = x - 2π

General Formulas and Concepts:

Pre-Algebra

• Order of Operations: BPEMDAS

Algebra I

Point-Slope Form: y - y₁ = m(x - x₁)

• x₁ - x coordinate

• y₁ - y coordinate

• m - slope

Pre-Calculus

• Unit Circle

Calculus

The definition of a derivative is the slope of the tangent line.

Derivatives of Trig Functions

• sin(x) = cos(x)
• cos(x) = -sin(x)
• tan(x) = sec²(x)
• sec(x) = sec(x)tan(x)
• csc(x) = -csc(x)cot(x)
• cot(x) = -csc²(x)

Explanation:

Step 1: Define

y = sinx

x = 2π

Step 2: Find Derivative

1. Take derivative: y' = cosx

This is the function of the tangent line for y.

Step 3: Find slope

1. Substitute in x into y': y'(2π) = cos2π
2. Evaluate: y'(2π) = 1

This tells us that the slope of the tangent line is 1 at x = 2π.

Step 4: Write equation

Find a point.

1. Substitute x into y: y(2π) = sin2π
2. Evaluate: y(2π) = 0
3. Write coordinate: (2π, 0)

Write tangent function.

1. Substitute: y - 0 = 1(x - 2π)
2. Simplify: y = x - 2π

This is the equation of the tangent line at x = 2π.