Basic Math Examples

100^{2} + {(\frac{h}{tan (22)})}^{2} = {(\frac{h}{tan (19)})}^{2}

$100^{2} + {(\frac{h}{\tan (22)})}^{2} = {(\frac{h}{\tan (19)})}^{2}$

Step 1

Subtract

{(\frac{h}{tan (19)})}^{2}

${(\frac{h}{\tan (19)})}^{2}$ from both sides of the equation.

100^{2} + {(\frac{h}{tan (22)})}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$100^{2} + {(\frac{h}{\tan (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2

Simplify the left side of the equation.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Raise

100

$100$ to the power of

2

$2$ .

10000 + {(\frac{h}{tan (22)})}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(\frac{h}{\tan (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.2

Separate fractions.

10000 + {(\frac{h}{1} \cdot \frac{1}{tan (22)})}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(\frac{h}{1} \cdot \frac{1}{\tan (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.3

Rewrite

tan (22)

$\tan (22)$ in terms of sines and cosines.

10000 + {⎛ ⎜ ⎝ \frac{h}{1} \cdot \frac{1}{\frac{sin (22)}{cos (22)}} ⎞ ⎟ ⎠}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(\frac{h}{1} \cdot \frac{1}{\frac{\sin (22)}{\cos (22)}})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.4

Multiply by the reciprocal of the fraction to divide by

\frac{sin (22)}{cos (22)}

$\frac{\sin (22)}{\cos (22)}$ .

10000 + {(\frac{h}{1} \cdot \frac{cos (22)}{sin (22)})}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(\frac{h}{1} \cdot \frac{\cos (22)}{\sin (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.5

Convert from

\frac{cos (22)}{sin (22)}

$\frac{\cos (22)}{\sin (22)}$ to

cot (22)

$\cot (22)$ .

10000 + {(\frac{h}{1} \cdot cot (22))}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(\frac{h}{1} \cdot \cot (22))}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.6

Divide

h

$h$ by

1

$1$ .

10000 + {(h cot (22))}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(h \cot (22))}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.7

Evaluate

cot (22)

$\cot (22)$ .

10000 + {(h \cdot 2.47508685)}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(h \cdot 2.47508685)}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.8

Move

2.47508685

$2.47508685$ to the left of

h

$h$ .

10000 + {(2.47508685 \cdot h)}^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {(2.47508685 \cdot h)}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.9

Apply the product rule to

2.47508685 h

$2.47508685 h$ .

10000 + {2.47508685}^{2} h^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + {2.47508685}^{2} h^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.10

Raise

2.47508685

$2.47508685$ to the power of

2

$2$ .

10000 + 6.12605493 h^{2} - {(\frac{h}{tan (19)})}^{2} = 0

$10000 + 6.12605493 h^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.11

Separate fractions.

10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{1}{tan (19)})}^{2} = 0

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{1}{\tan (19)})}^{2} = 0$

Step 2.1.12

Rewrite

tan (19)

$\tan (19)$ in terms of sines and cosines.

10000 + 6.12605493 h^{2} - {⎛ ⎜ ⎝ \frac{h}{1} \cdot \frac{1}{\frac{sin (19)}{cos (19)}} ⎞ ⎟ ⎠}^{2} = 0

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{1}{\frac{\sin (19)}{\cos (19)}})}^{2} = 0$

Step 2.1.13

Multiply by the reciprocal of the fraction to divide by

\frac{sin (19)}{cos (19)}

$\frac{\sin (19)}{\cos (19)}$ .

10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{cos (19)}{sin (19)})}^{2} = 0

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{\cos (19)}{\sin (19)})}^{2} = 0$

Step 2.1.14

Convert from

\frac{cos (19)}{sin (19)}

$\frac{\cos (19)}{\sin (19)}$ to

cot (19)

$\cot (19)$ .

10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot cot (19))}^{2} = 0

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \cot (19))}^{2} = 0$

Step 2.1.15

Divide

h

$h$ by

1

$1$ .

10000 + 6.12605493 h^{2} - {(h cot (19))}^{2} = 0

$10000 + 6.12605493 h^{2} - {(h \cot (19))}^{2} = 0$

Step 2.1.16

Evaluate

cot (19)

$\cot (19)$ .

10000 + 6.12605493 h^{2} - {(h \cdot 2.90421087)}^{2} = 0

$10000 + 6.12605493 h^{2} - {(h \cdot 2.90421087)}^{2} = 0$

Step 2.1.17

Move

2.90421087

$2.90421087$ to the left of

h

$h$ .

10000 + 6.12605493 h^{2} - {(2.90421087 \cdot h)}^{2} = 0

$10000 + 6.12605493 h^{2} - {(2.90421087 \cdot h)}^{2} = 0$

Step 2.1.18

Apply the product rule to

2.90421087 h

$2.90421087 h$ .

10000 + 6.12605493 h^{2} - ({2.90421087}^{2} h^{2}) = 0

$10000 + 6.12605493 h^{2} - ({2.90421087}^{2} h^{2}) = 0$

Step 2.1.19

Raise

2.90421087

$2.90421087$ to the power of

2

$2$ .

10000 + 6.12605493 h^{2} - (8.43444082 h^{2}) = 0

$10000 + 6.12605493 h^{2} - (8.43444082 h^{2}) = 0$

Step 2.1.20

Multiply

8.43444082

$8.43444082$ by

- 1

$- 1$ .

10000 + 6.12605493 h^{2} - 8.43444082 h^{2} = 0

$10000 + 6.12605493 h^{2} - 8.43444082 h^{2} = 0$

10000 + 6.12605493 h^{2} - 8.43444082 h^{2} = 0

$10000 + 6.12605493 h^{2} - 8.43444082 h^{2} = 0$

Step 2.2

Subtract

8.43444082 h^{2}

$8.43444082 h^{2}$ from

6.12605493 h^{2}

$6.12605493 h^{2}$ .

10000 - 2.30838589 h^{2} = 0

$10000 - 2.30838589 h^{2} = 0$

10000 - 2.30838589 h^{2} = 0

$10000 - 2.30838589 h^{2} = 0$

Step 3

Factor

10000 - 2.30838589 h^{2}

$10000 - 2.30838589 h^{2}$ .

Tap for more steps...

Step 3.1

Rewrite

10000

$10000$ as

100^{2}

$100^{2}$ .

100^{2} - 2.30838589 h^{2} = 0

$100^{2} - 2.30838589 h^{2} = 0$

Step 3.2

Rewrite

2.30838589 h^{2}

$2.30838589 h^{2}$ as

{(1.51933731 h)}^{2}

${(1.51933731 h)}^{2}$ .

100^{2} - {(1.51933731 h)}^{2} = 0

$100^{2} - {(1.51933731 h)}^{2} = 0$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 100

$a = 100$ and

b = 1.51933731 h

$b = 1.51933731 h$ .

(100 + 1.51933731 h) (100 - (1.51933731 h)) = 0

$(100 + 1.51933731 h) (100 - (1.51933731 h)) = 0$

Step 3.4

Multiply

1.51933731

$1.51933731$ by

- 1

$- 1$ .

(100 + 1.51933731 h) (100 - 1.51933731 h) = 0

$(100 + 1.51933731 h) (100 - 1.51933731 h) = 0$

(100 + 1.51933731 h) (100 - 1.51933731 h) = 0

$(100 + 1.51933731 h) (100 - 1.51933731 h) = 0$

Step 4

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

100 + 1.51933731 h = 0

$100 + 1.51933731 h = 0$

100 - 1.51933731 h = 0

$100 - 1.51933731 h = 0$

Step 5

Set

100 + 1.51933731 h

$100 + 1.51933731 h$ equal to

0

$0$ and solve for

h

$h$ .

Tap for more steps...

Step 5.1

Set

100 + 1.51933731 h

$100 + 1.51933731 h$ equal to

0

$0$ .

100 + 1.51933731 h = 0

$100 + 1.51933731 h = 0$

Step 5.2

Solve

100 + 1.51933731 h = 0

$100 + 1.51933731 h = 0$ for

h

$h$ .

Tap for more steps...

Step 5.2.1

Subtract

100

$100$ from both sides of the equation.

1.51933731 h = - 100

$1.51933731 h = - 100$

Step 5.2.2

Divide each term in

1.51933731 h = - 100

$1.51933731 h = - 100$ by

1.51933731

$1.51933731$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in

1.51933731 h = - 100

$1.51933731 h = - 100$ by

1.51933731

$1.51933731$ .

\frac{1.51933731 h}{1.51933731} = \frac{- 100}{1.51933731}

$\frac{1.51933731 h}{1.51933731} = \frac{- 100}{1.51933731}$

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of

1.51933731

$1.51933731$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

\frac{1.51933731 h}{1.51933731} = \frac{- 100}{1.51933731}

$\frac{1.51933731 h}{1.51933731} = \frac{- 100}{1.51933731}$

Step 5.2.2.2.1.2

Divide

$h$ by

$1$ .

$h = \frac{- 100}{1.51933731}$

Step 5.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.2.3.1

Divide

$- 100$ by

$1.51933731$ .

$h = - 65.8181687$

Step 6

Set

$100 - 1.51933731 h$ equal to

$0$ and solve for

$h$ .

Tap for more steps...

Step 6.1

Set

$100 - 1.51933731 h$ equal to

$0$ .

$100 - 1.51933731 h = 0$

Step 6.2

Solve

$100 - 1.51933731 h = 0$ for

$h$ .

Tap for more steps...

Step 6.2.1

Subtract

$100$ from both sides of the equation.

$- 1.51933731 h = - 100$

Step 6.2.2

Divide each term in

$- 1.51933731 h = - 100$ by

$- 1.51933731$ and simplify.

Tap for more steps...

Step 6.2.2.1

Divide each term in

$- 1.51933731 h = - 100$ by

$- 1.51933731$ .

$\frac{- 1.51933731 h}{- 1.51933731} = \frac{- 100}{- 1.51933731}$

Step 6.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.2.1

Cancel the common factor of

$- 1.51933731$ .

Tap for more steps...

Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{- 1.51933731 h}{- 1.51933731} = \frac{- 100}{- 1.51933731}$

Step 6.2.2.2.1.2

Divide

$h$ by

$1$ .

$h = \frac{- 100}{- 1.51933731}$

Step 6.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.2.3.1

Divide

$- 100$ by

$- 1.51933731$ .

$h = 65.8181687$

Step 7

The final solution is all the values that make

$(100 + 1.51933731 h) (100 - 1.51933731 h) = 0$ true.

$h = - 65.8181687, 65.8181687$