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

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

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Thành Đạt 3 years 2021-08-01T02:08:46+00:00 1 Answers 3 views 0

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  1. ngocdiep
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    2021-08-01T02:09:46+00:00
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    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

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