Calculus Examples

5 \cos (x) - 4 = 0

$5 \cos (x) - 4 = 0$

Step 1

Add

4

$4$ to both sides of the equation.

5 \cos (x) = 4

$5 \cos (x) = 4$

Step 2

Divide each term in

5 \cos (x) = 4

$5 \cos (x) = 4$ by

5

$5$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in

5 \cos (x) = 4

$5 \cos (x) = 4$ by

5

$5$ .

\frac{5 \cos (x)}{5} = \frac{4}{5}

$\frac{5 \cos (x)}{5} = \frac{4}{5}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of

5

$5$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

\frac{5 \cos (x)}{5} = \frac{4}{5}

$\frac{5 \cos (x)}{5} = \frac{4}{5}$

Step 2.2.1.2

Divide

\cos (x)

$\cos (x)$ by

1

$1$ .

\cos (x) = \frac{4}{5}

$\cos (x) = \frac{4}{5}$

\cos (x) = \frac{4}{5}

$\cos (x) = \frac{4}{5}$

\cos (x) = \frac{4}{5}

$\cos (x) = \frac{4}{5}$

\cos (x) = \frac{4}{5}

$\cos (x) = \frac{4}{5}$

Step 3

Take the inverse cosine of both sides of the equation to extract

x

$x$ from inside the cosine.

x = \arccos (\frac{4}{5})

$x = \arccos (\frac{4}{5})$

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

Evaluate

\arccos (\frac{4}{5})

$\arccos (\frac{4}{5})$ .

x = 0.6435011

$x = 0.6435011$

x = 0.6435011

$x = 0.6435011$

Step 5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

2 π

$2 π$ to find the solution in the fourth quadrant.

x = 2 (3.14159265) - 0.6435011

$x = 2 (3.14159265) - 0.6435011$

Step 6

Simplify

2 (3.14159265) - 0.6435011

$2 (3.14159265) - 0.6435011$ .

Tap for more steps...

Step 6.1

Multiply

2

$2$ by

3.14159265

$3.14159265$ .

x = 6.2831853 - 0.6435011

$x = 6.2831853 - 0.6435011$

Step 6.2

Subtract

0.6435011

$0.6435011$ from

6.2831853

$6.2831853$ .

x = 5.63968419

$x = 5.63968419$

x = 5.63968419

$x = 5.63968419$

Step 7

Find the period of

\cos (x)

$\cos (x)$ .

Tap for more steps...

Step 7.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 7.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 7.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 7.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 8

The period of the

\cos (x)

$\cos (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = 0.6435011 + 2 π n, 5.63968419 + 2 π n

$x = 0.6435011 + 2 π n, 5.63968419 + 2 π n$ , for any integer

n

$n$