Question

Find the equation of the line tangent to y = sin(x) going through х = pi/4​

Answer 1

`\displaystyle y - (√(2))/(2) = (√(2))/(2) \bigg( x - (\pi)/(4) \bigg)`

General Formulas and Concepts:

Algebra I

Coordinates (x, y)

Functions

Function Notation

Point-Slope Form: y - y₁ = m(x - x₁)

• x₁ - x coordinate
• y₁ - y coordinate
• m - slope

Pre-Calculus

• Unit Circle

Calculus

Derivatives

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

Derivative Notation

Trig Derivative:
`\displaystyle (d)/(dx)[sin(u)] = u'cos(u)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = sin(x)`

`\displaystyle x = (\pi)/(4)`

Step 2: Differentiate

1. Trig Derivative:
   `\displaystyle y' = cos(x)`

Step 3: Find Tangent Slope

1. Substitute in x [Derivative]:
   `\displaystyle y' \bigg( (\pi)/(4) \bigg) = cos \bigg( (\pi)/(4) \bigg)`
2. Evaluate [Unit Circle]:
   `\displaystyle y' \bigg( (\pi)/(4) \bigg) = (√(2))/(2)`

Step 4: Find Tangent Equation

1. Substitute in x [Function y]:
   `\displaystyle y \bigg( (\pi)/(4) \bigg) = sin \bigg( (\pi)/(4) \bigg)`
2. Evaluate [Unit Circle]:
   `\displaystyle y \bigg( (\pi)/(4) \bigg) = (√(2))/(2)`
3. Substitute in variables [Point-Slope Form]:
   `\displaystyle y - (√(2))/(2) = (√(2))/(2) \bigg( x - (\pi)/(4) \bigg)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Derivatives

Book: College Calculus 10e