Calculus Examples

$y = \tan (x)$ ; $x = π$

Step 1

Find the corresponding $y$ -value to $x = π$ .

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Step 1.1

Substitute $π$ in for $x$ .

$y = \tan (π)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \tan (π)$

Step 1.2.2

Simplify $\tan (π)$ .

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Step 1.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$y = - \tan (0)$

Step 1.2.2.2

The exact value of $\tan (0)$ is $0$ .

$y = - 0$

Step 1.2.2.3

Multiply $- 1$ by $0$ .

$y = 0$

Step 2

Find the first derivative and evaluate at $x = π$ and $y = 0$ to find the slope of the tangent line.

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Step 2.1

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 2.2

Evaluate the derivative at $x = π$ .

$\sec^{2} (π)$

Step 2.3

Simplify.

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Step 2.3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the second quadrant.

${(- \sec (0))}^{2}$

Step 2.3.2

The exact value of $\sec (0)$ is $1$ .

${(- 1 \cdot 1)}^{2}$

Step 2.3.3

Multiply $- 1$ by $1$ .

${(- 1)}^{2}$

Step 2.3.4

Raise $- 1$ to the power of $2$ .

$1$

Step 3

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

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Step 3.1

Use the slope $1$ and a given point $(π, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 1 \cdot (x - (π))$

Step 3.2

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

$y + 0 = 1 \cdot (x - π)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Add $y$ and $0$ .

$y = 1 \cdot (x - π)$

Step 3.3.2

Multiply $x - π$ by $1$ .

$y = x - π$

Step 4