Trigonometry Examples

$(\frac{1}{5} z^{2} - \frac{1}{3} z) - (- \frac{7}{15} z^{2} + \frac{1}{6} z - \frac{14}{15})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Combine $\frac{1}{5}$ and $z^{2}$ .

$\frac{z^{2}}{5} - \frac{1}{3} z - (- \frac{7}{15} z^{2} + \frac{1}{6} z - \frac{14}{15})$

Step 1.2

Combine $z$ and $\frac{1}{3}$ .

$\frac{z^{2}}{5} - \frac{z}{3} - (- \frac{7}{15} z^{2} + \frac{1}{6} z - \frac{14}{15})$

Step 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

Combine $z^{2}$ and $\frac{7}{15}$ .

$\frac{z^{2}}{5} - \frac{z}{3} - (- \frac{z^{2} \cdot 7}{15} + \frac{1}{6} z - \frac{14}{15})$

Step 1.3.2

Move $7$ to the left of $z^{2}$ .

$\frac{z^{2}}{5} - \frac{z}{3} - (- \frac{7 \cdot z^{2}}{15} + \frac{1}{6} z - \frac{14}{15})$

Step 1.3.3

Combine $\frac{1}{6}$ and $z$ .

$\frac{z^{2}}{5} - \frac{z}{3} - (- \frac{7 z^{2}}{15} + \frac{z}{6} - \frac{14}{15})$

Step 1.4

Apply the distributive property.

$\frac{z^{2}}{5} - \frac{z}{3} - - \frac{7 z^{2}}{15} - \frac{z}{6} - - \frac{14}{15}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Multiply $- - \frac{7 z^{2}}{15}$ .

Tap for more steps...

Step 1.5.1.1

Multiply $- 1$ by $- 1$ .

$\frac{z^{2}}{5} - \frac{z}{3} + 1 \frac{7 z^{2}}{15} - \frac{z}{6} - - \frac{14}{15}$

Step 1.5.1.2

Multiply $\frac{7 z^{2}}{15}$ by $1$ .

$\frac{z^{2}}{5} - \frac{z}{3} + \frac{7 z^{2}}{15} - \frac{z}{6} - - \frac{14}{15}$

Step 1.5.2

Multiply $- - \frac{14}{15}$ .

Tap for more steps...

Step 1.5.2.1

Multiply $- 1$ by $- 1$ .

$\frac{z^{2}}{5} - \frac{z}{3} + \frac{7 z^{2}}{15} - \frac{z}{6} + 1 (\frac{14}{15})$

Step 1.5.2.2

Multiply $\frac{14}{15}$ by $1$ .

$\frac{z^{2}}{5} - \frac{z}{3} + \frac{7 z^{2}}{15} - \frac{z}{6} + \frac{14}{15}$

Step 2

To write $\frac{z^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{z}{3} + \frac{z^{2}}{5} \cdot \frac{3}{3} + \frac{7 z^{2}}{15} - \frac{z}{6} + \frac{14}{15}$

Step 3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{z^{2}}{5}$ by $\frac{3}{3}$ .

$- \frac{z}{3} + \frac{z^{2} \cdot 3}{5 \cdot 3} + \frac{7 z^{2}}{15} - \frac{z}{6} + \frac{14}{15}$

Step 3.2

Multiply $5$ by $3$ .

$- \frac{z}{3} + \frac{z^{2} \cdot 3}{15} + \frac{7 z^{2}}{15} - \frac{z}{6} + \frac{14}{15}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Combine the numerators over the common denominator.

$- \frac{z}{3} + \frac{z^{2} \cdot 3 + 7 z^{2}}{15} - \frac{z}{6} + \frac{14}{15}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{- z}{3} + \frac{z^{2} \cdot 3 + 7 z^{2} + 14}{15} + \frac{- z}{6}$

Step 5

Move $3$ to the left of $z^{2}$ .

$\frac{- z}{3} + \frac{3 z^{2} + 7 z^{2} + 14}{15} + \frac{- z}{6}$

Step 6

Add $3 z^{2}$ and $7 z^{2}$ .

$\frac{- z}{3} + \frac{10 z^{2} + 14}{15} + \frac{- z}{6}$

Step 7

Find the common denominator.

Tap for more steps...

Step 7.1

Multiply $\frac{- z}{3}$ by $\frac{10}{10}$ .

$\frac{- z}{3} \cdot \frac{10}{10} + \frac{10 z^{2} + 14}{15} + \frac{- z}{6}$

Step 7.2

Multiply $\frac{- z}{3}$ by $\frac{10}{10}$ .

$\frac{- z \cdot 10}{3 \cdot 10} + \frac{10 z^{2} + 14}{15} + \frac{- z}{6}$

Step 7.3

Multiply $\frac{10 z^{2} + 14}{15}$ by $\frac{2}{2}$ .

$\frac{- z \cdot 10}{3 \cdot 10} + \frac{10 z^{2} + 14}{15} \cdot \frac{2}{2} + \frac{- z}{6}$

Step 7.4

Multiply $\frac{10 z^{2} + 14}{15}$ by $\frac{2}{2}$ .

$\frac{- z \cdot 10}{3 \cdot 10} + \frac{(10 z^{2} + 14) \cdot 2}{15 \cdot 2} + \frac{- z}{6}$

Step 7.5

Multiply $\frac{- z}{6}$ by $\frac{5}{5}$ .

$\frac{- z \cdot 10}{3 \cdot 10} + \frac{(10 z^{2} + 14) \cdot 2}{15 \cdot 2} + \frac{- z}{6} \cdot \frac{5}{5}$

Step 7.6

Multiply $\frac{- z}{6}$ by $\frac{5}{5}$ .

$\frac{- z \cdot 10}{3 \cdot 10} + \frac{(10 z^{2} + 14) \cdot 2}{15 \cdot 2} + \frac{- z \cdot 5}{6 \cdot 5}$

Step 7.7

Multiply $3$ by $10$ .

$\frac{- z \cdot 10}{30} + \frac{(10 z^{2} + 14) \cdot 2}{15 \cdot 2} + \frac{- z \cdot 5}{6 \cdot 5}$

Step 7.8

Reorder the factors of $15 \cdot 2$ .

$\frac{- z \cdot 10}{30} + \frac{(10 z^{2} + 14) \cdot 2}{2 \cdot 15} + \frac{- z \cdot 5}{6 \cdot 5}$

Step 7.9

Multiply $2$ by $15$ .

$\frac{- z \cdot 10}{30} + \frac{(10 z^{2} + 14) \cdot 2}{30} + \frac{- z \cdot 5}{6 \cdot 5}$

Step 7.10

Reorder the factors of $6 \cdot 5$ .

$\frac{- z \cdot 10}{30} + \frac{(10 z^{2} + 14) \cdot 2}{30} + \frac{- z \cdot 5}{5 \cdot 6}$

Step 7.11

Multiply $5$ by $6$ .

$\frac{- z \cdot 10}{30} + \frac{(10 z^{2} + 14) \cdot 2}{30} + \frac{- z \cdot 5}{30}$

Step 8

Combine the numerators over the common denominator.

$\frac{- z \cdot 10 + (10 z^{2} + 14) \cdot 2 - z \cdot 5}{30}$

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

Multiply $10$ by $- 1$ .

$\frac{- 10 z + (10 z^{2} + 14) \cdot 2 - z \cdot 5}{30}$

Step 9.2

Apply the distributive property.

$\frac{- 10 z + 10 z^{2} \cdot 2 + 14 \cdot 2 - z \cdot 5}{30}$

Step 9.3

Multiply $2$ by $10$ .

$\frac{- 10 z + 20 z^{2} + 14 \cdot 2 - z \cdot 5}{30}$

Step 9.4

Multiply $14$ by $2$ .

$\frac{- 10 z + 20 z^{2} + 28 - z \cdot 5}{30}$

Step 9.5

Multiply $5$ by $- 1$ .

$\frac{- 10 z + 20 z^{2} + 28 - 5 z}{30}$

Step 10

Simplify by adding terms.

Tap for more steps...

Step 10.1

Subtract $5 z$ from $- 10 z$ .

$\frac{- 15 z + 20 z^{2} + 28}{30}$

Step 10.2

Reorder terms.

$\frac{20 z^{2} - 15 z + 28}{30}$