Trigonometry Examples

${(10 \sqrt{3})}^{2} = {(15 \sqrt{3})}^{2} + {(6 \sqrt{3})}^{2} - 2 (15 \sqrt{3}) (6 \sqrt{3}) \cos (c)$

Step 1

Rewrite the equation as ${(15 \sqrt{3})}^{2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$ .

${(15 \sqrt{3})}^{2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2

Simplify ${(15 \sqrt{3})}^{2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $15 \sqrt{3}$ .

$15^{2} {\sqrt{3}}^{2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.2

Raise $15$ to the power of $2$ .

$225 {\sqrt{3}}^{2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$225 {(3^{\frac{1}{2}})}^{2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$225 \cdot 3^{\frac{1}{2} \cdot 2} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$225 \cdot 3^{\frac{2}{2}} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

$225 \cdot 3^{\frac{2}{2}} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.3.4.2

Rewrite the expression.

$225 \cdot 3^{1} + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.3.5

Evaluate the exponent.

$225 \cdot 3 + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.4

Multiply $225$ by $3$ .

$675 + {(6 \sqrt{3})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.5

Apply the product rule to $6 \sqrt{3}$ .

$675 + 6^{2} {\sqrt{3}}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.6

Raise $6$ to the power of $2$ .

$675 + 36 {\sqrt{3}}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$675 + 36 {(3^{\frac{1}{2}})}^{2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$675 + 36 \cdot 3^{\frac{1}{2} \cdot 2} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$675 + 36 \cdot 3^{\frac{2}{2}} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$675 + 36 \cdot 3^{\frac{2}{2}} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.7.4.2

Rewrite the expression.

$675 + 36 \cdot 3^{1} - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.7.5

Evaluate the exponent.

$675 + 36 \cdot 3 - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.8

Multiply $36$ by $3$ .

$675 + 108 - 2 \cdot (15 \sqrt{3}) \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.9

Multiply $15$ by $- 2$ .

$675 + 108 - 30 \sqrt{3} \cdot (6 \sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.10

Multiply $- 30 \sqrt{3} (6 \sqrt{3} \cos (c))$ .

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Step 2.1.10.1

Multiply $6$ by $- 30$ .

$675 + 108 - 180 \sqrt{3} (\sqrt{3} \cos (c)) = {(10 \sqrt{3})}^{2}$

Step 2.1.10.2

Raise $\sqrt{3}$ to the power of $1$ .

$675 + 108 - 180 ({\sqrt{3}}^{1} \sqrt{3}) \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.10.3

Raise $\sqrt{3}$ to the power of $1$ .

$675 + 108 - 180 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.10.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$675 + 108 - 180 {\sqrt{3}}^{1 + 1} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.10.5

Add $1$ and $1$ .

$675 + 108 - 180 {\sqrt{3}}^{2} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.11

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$675 + 108 - 180 {(3^{\frac{1}{2}})}^{2} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.11.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$675 + 108 - 180 \cdot 3^{\frac{1}{2} \cdot 2} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.11.3

Combine $\frac{1}{2}$ and $2$ .

$675 + 108 - 180 \cdot 3^{\frac{2}{2}} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.11.4

Cancel the common factor of $2$ .

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Step 2.1.11.4.1

Cancel the common factor.

$675 + 108 - 180 \cdot 3^{\frac{2}{2}} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.11.4.2

Rewrite the expression.

$675 + 108 - 180 \cdot 3^{1} \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.11.5

Evaluate the exponent.

$675 + 108 - 180 \cdot 3 \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.1.12

Multiply $- 180$ by $3$ .

$675 + 108 - 540 \cos (c) = {(10 \sqrt{3})}^{2}$

Step 2.2

Add $675$ and $108$ .

$783 - 540 \cos (c) = {(10 \sqrt{3})}^{2}$

Step 3

Simplify ${(10 \sqrt{3})}^{2}$ .

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Step 3.1

Simplify the expression.

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Step 3.1.1

Apply the product rule to $10 \sqrt{3}$ .

$783 - 540 \cos (c) = 10^{2} {\sqrt{3}}^{2}$

Step 3.1.2

Raise $10$ to the power of $2$ .

$783 - 540 \cos (c) = 100 {\sqrt{3}}^{2}$

Step 3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$783 - 540 \cos (c) = 100 {(3^{\frac{1}{2}})}^{2}$

Step 3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$783 - 540 \cos (c) = 100 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.2.3

Combine $\frac{1}{2}$ and $2$ .

$783 - 540 \cos (c) = 100 \cdot 3^{\frac{2}{2}}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$783 - 540 \cos (c) = 100 \cdot 3^{\frac{2}{2}}$

Step 3.2.4.2

Rewrite the expression.

$783 - 540 \cos (c) = 100 \cdot 3^{1}$

Step 3.2.5

Evaluate the exponent.

$783 - 540 \cos (c) = 100 \cdot 3$

Step 3.3

Multiply $100$ by $3$ .

$783 - 540 \cos (c) = 300$

Step 4

Move all terms not containing $\cos (c)$ to the right side of the equation.

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Step 4.1

Subtract $783$ from both sides of the equation.

$- 540 \cos (c) = 300 - 783$

Step 4.2

Subtract $783$ from $300$ .

$- 540 \cos (c) = - 483$

Step 5

Divide each term in $- 540 \cos (c) = - 483$ by $- 540$ and simplify.

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Step 5.1

Divide each term in $- 540 \cos (c) = - 483$ by $- 540$ .

$\frac{- 540 \cos (c)}{- 540} = \frac{- 483}{- 540}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 540$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 540 \cos (c)}{- 540} = \frac{- 483}{- 540}$

Step 5.2.1.2

Divide $\cos (c)$ by $1$ .

$\cos (c) = \frac{- 483}{- 540}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 483$ and $- 540$ .

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Step 5.3.1.1

Factor $- 3$ out of $- 483$ .

$\cos (c) = \frac{- 3 (161)}{- 540}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $- 3$ out of $- 540$ .

$\cos (c) = \frac{- 3 \cdot 161}{- 3 \cdot 180}$

Step 5.3.1.2.2

Cancel the common factor.

$\cos (c) = \frac{- 3 \cdot 161}{- 3 \cdot 180}$

Step 5.3.1.2.3

Rewrite the expression.

$\cos (c) = \frac{161}{180}$

Step 6

Take the inverse cosine of both sides of the equation to extract $c$ from inside the cosine.

$c = \arccos (\frac{161}{180})$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arccos (\frac{161}{180})$ .

$c = 0.46360902$

Step 8

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$c = 2 (3.14159265) - 0.46360902$

Step 9

Simplify $2 (3.14159265) - 0.46360902$ .

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Step 9.1

Multiply $2$ by $3.14159265$ .

$c = 6.2831853 - 0.46360902$

Step 9.2

Subtract $0.46360902$ from $6.2831853$ .

$c = 5.81957627$

Step 10

Find the period of $\cos (c)$ .

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

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

$\frac{2 π}{| 1 |}$

Step 10.3

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

$\frac{2 π}{1}$

Step 10.4

Divide $2 π$ by $1$ .

$2 π$

Step 11

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

$c = 0.46360902 + 2 π n, 5.81957627 + 2 π n$ , for any integer $n$