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Sagot :
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
- Trig Derivative: [tex]\displaystyle y' = cos(x)[/tex]
Step 3: Find Tangent Slope
- Substitute in x [Derivative]: [tex]\displaystyle y' \bigg( \frac{\pi}{4} \bigg) = cos \bigg( \frac{\pi}{4} \bigg)[/tex]
- Evaluate [Unit Circle]: [tex]\displaystyle y' \bigg( \frac{\pi}{4} \bigg) = \frac{\sqrt{2}}{2}[/tex]
Step 4: Find Tangent Equation
- Substitute in x [Function y]: [tex]\displaystyle y \bigg( \frac{\pi}{4} \bigg) = sin \bigg( \frac{\pi}{4} \bigg)[/tex]
- Evaluate [Unit Circle]: [tex]\displaystyle y \bigg( \frac{\pi}{4} \bigg) = \frac{\sqrt{2}}{2}[/tex]
- 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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