Calculus Examples

$y = \sin (x)$ , $(\frac{5 π}{2}, 1)$

Step 1

Find the first derivative and evaluate at $x = \frac{5 π}{2}$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 1.2

Evaluate the derivative at $x = \frac{5 π}{2}$ .

$\cos (\frac{5 π}{2})$

Step 1.3

Simplify.

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Step 1.3.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (\frac{π}{2})$

Step 1.3.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$0$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $0$ and a given point $(\frac{5 π}{2}, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 0 \cdot (x - (\frac{5 π}{2}))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 1 = 0 \cdot (x - \frac{5 π}{2})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $0$ by $x - \frac{5 π}{2}$ .

$y - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = 1$

Step 3