Question

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

Answer 1

Answer:

$\displaystyle y - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \bigg( x - \frac{\pi}{4} \bigg)$

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = sin(x)$

$\displaystyle x = \frac{\pi}{4}$

Step 2: Differentiate

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

Step 3: Find Tangent Slope

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

Step 4: Find Tangent Equation

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

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

Unit: Derivatives

Book: College Calculus 10e