Trigonometry Examples

$\frac{1}{3} \cdot (d - 4) - \frac{3}{2} \cdot (3 d - 3) = \frac{7}{12}$

Step 1

Simplify $\frac{1}{3} \cdot (d - 4) - \frac{3}{2} \cdot (3 d - 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{1}{3} d + \frac{1}{3} \cdot - 4 - \frac{3}{2} \cdot (3 d - 3) = \frac{7}{12}$

Step 1.1.2

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

$\frac{d}{3} + \frac{1}{3} \cdot - 4 - \frac{3}{2} \cdot (3 d - 3) = \frac{7}{12}$

Step 1.1.3

Combine $\frac{1}{3}$ and $- 4$ .

$\frac{d}{3} + \frac{- 4}{3} - \frac{3}{2} \cdot (3 d - 3) = \frac{7}{12}$

Step 1.1.4

Move the negative in front of the fraction.

$\frac{d}{3} - \frac{4}{3} - \frac{3}{2} \cdot (3 d - 3) = \frac{7}{12}$

Step 1.1.5

Apply the distributive property.

$\frac{d}{3} - \frac{4}{3} - \frac{3}{2} (3 d) - \frac{3}{2} \cdot - 3 = \frac{7}{12}$

Step 1.1.6

Multiply $- \frac{3}{2} (3 d)$ .

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Step 1.1.6.1

Multiply $3$ by $- 1$ .

$\frac{d}{3} - \frac{4}{3} - 3 (\frac{3}{2}) d - \frac{3}{2} \cdot - 3 = \frac{7}{12}$

Step 1.1.6.2

Combine $- 3$ and $\frac{3}{2}$ .

$\frac{d}{3} - \frac{4}{3} + \frac{- 3 \cdot 3}{2} d - \frac{3}{2} \cdot - 3 = \frac{7}{12}$

Step 1.1.6.3

Multiply $- 3$ by $3$ .

$\frac{d}{3} - \frac{4}{3} + \frac{- 9}{2} d - \frac{3}{2} \cdot - 3 = \frac{7}{12}$

Step 1.1.6.4

Combine $\frac{- 9}{2}$ and $d$ .

$\frac{d}{3} - \frac{4}{3} + \frac{- 9 d}{2} - \frac{3}{2} \cdot - 3 = \frac{7}{12}$

Step 1.1.7

Multiply $- \frac{3}{2} \cdot - 3$ .

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Step 1.1.7.1

Multiply $- 3$ by $- 1$ .

$\frac{d}{3} - \frac{4}{3} + \frac{- 9 d}{2} + 3 (\frac{3}{2}) = \frac{7}{12}$

Step 1.1.7.2

Combine $3$ and $\frac{3}{2}$ .

$\frac{d}{3} - \frac{4}{3} + \frac{- 9 d}{2} + \frac{3 \cdot 3}{2} = \frac{7}{12}$

Step 1.1.7.3

Multiply $3$ by $3$ .

$\frac{d}{3} - \frac{4}{3} + \frac{- 9 d}{2} + \frac{9}{2} = \frac{7}{12}$

Step 1.1.8

Move the negative in front of the fraction.

$\frac{d}{3} - \frac{4}{3} - \frac{9 d}{2} + \frac{9}{2} = \frac{7}{12}$

Step 1.2

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

$\frac{d}{3} \cdot \frac{2}{2} - \frac{9 d}{2} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.3

To write $- \frac{9 d}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{d}{3} \cdot \frac{2}{2} - \frac{9 d}{2} \cdot \frac{3}{3} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{d}{3}$ by $\frac{2}{2}$ .

$\frac{d \cdot 2}{3 \cdot 2} - \frac{9 d}{2} \cdot \frac{3}{3} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{d \cdot 2}{6} - \frac{9 d}{2} \cdot \frac{3}{3} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.4.3

Multiply $\frac{9 d}{2}$ by $\frac{3}{3}$ .

$\frac{d \cdot 2}{6} - \frac{9 d \cdot 3}{2 \cdot 3} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.4.4

Multiply $2$ by $3$ .

$\frac{d \cdot 2}{6} - \frac{9 d \cdot 3}{6} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4}{3} + \frac{9}{2} = \frac{7}{12}$

Step 1.6

Find the common denominator.

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Step 1.6.1

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - (\frac{4}{3} \cdot \frac{2}{2}) + \frac{9}{2} = \frac{7}{12}$

Step 1.6.2

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4 \cdot 2}{3 \cdot 2} + \frac{9}{2} = \frac{7}{12}$

Step 1.6.3

Multiply $\frac{9}{2}$ by $\frac{3}{3}$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4 \cdot 2}{3 \cdot 2} + \frac{9}{2} \cdot \frac{3}{3} = \frac{7}{12}$

Step 1.6.4

Multiply $\frac{9}{2}$ by $\frac{3}{3}$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4 \cdot 2}{3 \cdot 2} + \frac{9 \cdot 3}{2 \cdot 3} = \frac{7}{12}$

Step 1.6.5

Reorder the factors of $3 \cdot 2$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4 \cdot 2}{2 \cdot 3} + \frac{9 \cdot 3}{2 \cdot 3} = \frac{7}{12}$

Step 1.6.6

Multiply $2$ by $3$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4 \cdot 2}{6} + \frac{9 \cdot 3}{2 \cdot 3} = \frac{7}{12}$

Step 1.6.7

Multiply $2$ by $3$ .

$\frac{d \cdot 2 - 9 d \cdot 3}{6} - \frac{4 \cdot 2}{6} + \frac{9 \cdot 3}{6} = \frac{7}{12}$

Step 1.7

Combine the numerators over the common denominator.

$\frac{d \cdot 2 - 9 d \cdot 3 - 4 \cdot 2 + 9 \cdot 3}{6} = \frac{7}{12}$

Step 1.8

Simplify each term.

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Step 1.8.1

Move $2$ to the left of $d$ .

$\frac{2 \cdot d - 9 d \cdot 3 - 4 \cdot 2 + 9 \cdot 3}{6} = \frac{7}{12}$

Step 1.8.2

Multiply $3$ by $- 9$ .

$\frac{2 d - 27 d - 4 \cdot 2 + 9 \cdot 3}{6} = \frac{7}{12}$

Step 1.8.3

Multiply $- 4$ by $2$ .

$\frac{2 d - 27 d - 8 + 9 \cdot 3}{6} = \frac{7}{12}$

Step 1.8.4

Multiply $9$ by $3$ .

$\frac{2 d - 27 d - 8 + 27}{6} = \frac{7}{12}$

Step 1.9

Simplify terms.

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Step 1.9.1

Subtract $27 d$ from $2 d$ .

$\frac{- 25 d - 8 + 27}{6} = \frac{7}{12}$

Step 1.9.2

Add $- 8$ and $27$ .

$\frac{- 25 d + 19}{6} = \frac{7}{12}$

Step 1.9.3

Factor $- 1$ out of $- 25 d$ .

$\frac{- (25 d) + 19}{6} = \frac{7}{12}$

Step 1.9.4

Rewrite $19$ as $- 1 (- 19)$ .

$\frac{- (25 d) - 1 (- 19)}{6} = \frac{7}{12}$

Step 1.9.5

Factor $- 1$ out of $- (25 d) - 1 (- 19)$ .

$\frac{- (25 d - 19)}{6} = \frac{7}{12}$

Step 1.9.6

Simplify the expression.

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Step 1.9.6.1

Rewrite $- (25 d - 19)$ as $- 1 (25 d - 19)$ .

$\frac{- 1 (25 d - 19)}{6} = \frac{7}{12}$

Step 1.9.6.2

Move the negative in front of the fraction.

$- \frac{25 d - 19}{6} = \frac{7}{12}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$d \approx 0.61999999$

Step 3