AnnaDbar
Answered

Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

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

We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.