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

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Answer:

[tex]\displaystyle y - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \bigg( x - \frac{\pi}{4} \bigg)[/tex]

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:                                                                                                          [tex]\displaystyle \frac{d}{dx}[sin(u)] = u'cos(u)[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle y = sin(x)[/tex]

[tex]\displaystyle x = \frac{\pi}{4}[/tex]

Step 2: Differentiate

  1. Trig Derivative:                                                                                                 [tex]\displaystyle y' = cos(x)[/tex]

Step 3: Find Tangent Slope

  1. Substitute in x [Derivative]:                                                                              [tex]\displaystyle y' \bigg( \frac{\pi}{4} \bigg) = cos \bigg( \frac{\pi}{4} \bigg)[/tex]
  2. Evaluate [Unit Circle]:                                                                                       [tex]\displaystyle y' \bigg( \frac{\pi}{4} \bigg) = \frac{\sqrt{2}}{2}[/tex]

Step 4: Find Tangent Equation

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

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

Unit: Derivatives

Book: College Calculus 10e

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