Calculus Examples

$y = \cos (x)$

Step 1

Set $y$ as a function of $x$ .

$f (x) = \cos (x)$

Step 2

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

$- \sin (x)$

Step 3

Set the derivative equal to $0$ then solve the equation $- \sin (x) = 0$ .

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Step 3.1

Divide each term in $- \sin (x) = 0$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \sin (x) = 0$ by $- 1$ .

$\frac{- \sin (x)}{- 1} = \frac{0}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (x)}{1} = \frac{0}{- 1}$

Step 3.1.2.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{0}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $- 1$ .

$\sin (x) = 0$

Step 3.2

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (0)$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

$x = 0$

Step 3.4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Step 3.5

Subtract $0$ from $π$ .

$x = π$

Step 3.6

Find the period of $\sin (x)$ .

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Step 3.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.7

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 3.8

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 4

Solve the original function $f (x) = \cos (x)$ at $x = π$ .

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Step 4.1

$f (π) = \cos (π)$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (π) = - \cos (0)$

Step 4.2.2

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

$f (π) = - 1 \cdot 1$

Step 4.2.3

Multiply $- 1$ by $1$ .

$f (π) = - 1$

Step 4.2.4

The final answer is $- 1$ .

$- 1$

Step 5

The horizontal tangent line on function $f (x) = \cos (x)$ is $y = x = π n, y = - 1$ .

$y = x = π n, y = - 1$

Step 6