AnnaDbar
Answered

Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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

Sagot :

Space

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

Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.