Trigonometry Examples

$2 \cos (x) + 2 \sin (x) = \sqrt{6}$

Step 1

Square both sides of the equation.

${(2 \cos (x) + 2 \sin (x))}^{2} = {(\sqrt{6})}^{2}$

Step 2

Simplify ${(2 \cos (x) + 2 \sin (x))}^{2}$ .

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Step 2.1

Rewrite ${(2 \cos (x) + 2 \sin (x))}^{2}$ as $(2 \cos (x) + 2 \sin (x)) (2 \cos (x) + 2 \sin (x))$ .

$(2 \cos (x) + 2 \sin (x)) (2 \cos (x) + 2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.2

Expand $(2 \cos (x) + 2 \sin (x)) (2 \cos (x) + 2 \sin (x))$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$2 \cos (x) (2 \cos (x) + 2 \sin (x)) + 2 \sin (x) (2 \cos (x) + 2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.2.2

Apply the distributive property.

$2 \cos (x) (2 \cos (x)) + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x) + 2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.2.3

Apply the distributive property.

$2 \cos (x) (2 \cos (x)) + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $2 \cos (x) (2 \cos (x))$ .

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Step 2.3.1.1.1

Multiply $2$ by $2$ .

$4 \cos (x) \cos (x) + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.1.2

Raise $\cos (x)$ to the power of $1$ .

$4 (\cos^{1} (x) \cos (x)) + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.1.3

Raise $\cos (x)$ to the power of $1$ .

$4 (\cos^{1} (x) \cos^{1} (x)) + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.1.4

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

$4 {\cos (x)}^{1 + 1} + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.1.5

Add $1$ and $1$ .

$4 \cos^{2} (x) + 2 \cos (x) (2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.2

Add parentheses.

$4 \cos^{2} (x) + 2 (\cos (x) (2 \sin (x))) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.3

Reorder $\cos (x)$ and $2 \sin (x)$ .

$4 \cos^{2} (x) + 2 (2 \sin (x) \cos (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.4

Apply the sine double-angle identity.

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.5

Add parentheses.

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 (\sin (x) (2 \cos (x))) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.6

Reorder $\sin (x)$ and $2$ .

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 (2 \cdot \sin (x) \cos (x)) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.7

Apply the sine double-angle identity.

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (2 x) + 2 \sin (x) (2 \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.8

Multiply $2 \sin (x) (2 \sin (x))$ .

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Step 2.3.1.8.1

Multiply $2$ by $2$ .

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (2 x) + 4 \sin (x) \sin (x) = {(\sqrt{6})}^{2}$

Step 2.3.1.8.2

Raise $\sin (x)$ to the power of $1$ .

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (2 x) + 4 (\sin^{1} (x) \sin (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.8.3

Raise $\sin (x)$ to the power of $1$ .

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (2 x) + 4 (\sin^{1} (x) \sin^{1} (x)) = {(\sqrt{6})}^{2}$

Step 2.3.1.8.4

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

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (2 x) + 4 {\sin (x)}^{1 + 1} = {(\sqrt{6})}^{2}$

Step 2.3.1.8.5

Add $1$ and $1$ .

$4 \cos^{2} (x) + 2 \sin (2 x) + 2 \sin (2 x) + 4 \sin^{2} (x) = {(\sqrt{6})}^{2}$

Step 2.3.2

Add $2 \sin (2 x)$ and $2 \sin (2 x)$ .

$4 \cos^{2} (x) + 4 \sin (2 x) + 4 \sin^{2} (x) = {(\sqrt{6})}^{2}$

Step 2.4

Move $4 \sin^{2} (x)$ .

$4 \cos^{2} (x) + 4 \sin^{2} (x) + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 2.5

Factor $4$ out of $4 \cos^{2} (x)$ .

$4 (\cos^{2} (x)) + 4 \sin^{2} (x) + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 2.6

Factor $4$ out of $4 \sin^{2} (x)$ .

$4 (\cos^{2} (x)) + 4 (\sin^{2} (x)) + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 2.7

Factor $4$ out of $4 (\cos^{2} (x)) + 4 (\sin^{2} (x))$ .

$4 (\cos^{2} (x) + \sin^{2} (x)) + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 2.8

Rearrange terms.

$4 (\sin^{2} (x) + \cos^{2} (x)) + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 2.9

Apply pythagorean identity.

$4 \cdot 1 + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 2.10

Multiply $4$ by $1$ .

$4 + 4 \sin (2 x) = {(\sqrt{6})}^{2}$

Step 3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$4 + 4 \sin (2 x) = {(6^{\frac{1}{2}})}^{2}$

Step 3.2

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

$4 + 4 \sin (2 x) = 6^{\frac{1}{2} \cdot 2}$

Step 3.3

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

$4 + 4 \sin (2 x) = 6^{\frac{2}{2}}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$4 + 4 \sin (2 x) = 6^{\frac{2}{2}}$

Step 3.4.2

Rewrite the expression.

$4 + 4 \sin (2 x) = 6^{1}$

Step 3.5

Evaluate the exponent.

$4 + 4 \sin (2 x) = 6$

Step 4

Move all terms not containing $\sin (2 x)$ to the right side of the equation.

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Step 4.1

Subtract $4$ from both sides of the equation.

$4 \sin (2 x) = 6 - 4$

Step 4.2

Subtract $4$ from $6$ .

$4 \sin (2 x) = 2$

Step 5

Divide each term in $4 \sin (2 x) = 2$ by $4$ and simplify.

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Step 5.1

Divide each term in $4 \sin (2 x) = 2$ by $4$ .

$\frac{4 \sin (2 x)}{4} = \frac{2}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 \sin (2 x)}{4} = \frac{2}{4}$

Step 5.2.1.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{2}{4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $2$ and $4$ .

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Step 5.3.1.1

Factor $2$ out of $2$ .

$\sin (2 x) = \frac{2 (1)}{4}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $4$ .

$\sin (2 x) = \frac{2 \cdot 1}{2 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$\sin (2 x) = \frac{2 \cdot 1}{2 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

$\sin (2 x) = \frac{1}{2}$

Step 6

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$2 x = \arcsin (\frac{1}{2})$

Step 7

Simplify the right side.

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Step 7.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$2 x = \frac{π}{6}$

Step 8

Divide each term in $2 x = \frac{π}{6}$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 x = \frac{π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{2}$

Step 8.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{2}$ .

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Step 8.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{2}$ .

$x = \frac{π}{6 \cdot 2}$

Step 8.3.2.2

Multiply $6$ by $2$ .

$x = \frac{π}{12}$

Step 9

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

$2 x = π - \frac{π}{6}$

Step 10

Solve for $x$ .

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Step 10.1

Simplify.

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Step 10.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 10.1.2

Combine $π$ and $\frac{6}{6}$ .

$2 x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 10.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 6 - π}{6}$

Step 10.1.4

Subtract $π$ from $π \cdot 6$ .

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Step 10.1.4.1

Reorder $π$ and $6$ .

$2 x = \frac{6 \cdot π - π}{6}$

Step 10.1.4.2

Subtract $π$ from $6 \cdot π$ .

$2 x = \frac{5 \cdot π}{6}$

Step 10.2

Divide each term in $2 x = \frac{5 \cdot π}{6}$ by $2$ and simplify.

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Step 10.2.1

Divide each term in $2 x = \frac{5 \cdot π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}$

Step 10.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{6}}{2}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 \cdot π}{6} \cdot \frac{1}{2}$

Step 10.2.3.2

Multiply $\frac{5 π}{6} \cdot \frac{1}{2}$ .

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Step 10.2.3.2.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{6 \cdot 2}$

Step 10.2.3.2.2

Multiply $6$ by $2$ .

$x = \frac{5 π}{12}$

Step 11

Find the period of $\sin (2 x)$ .

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 11.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 11.4

Cancel the common factor of $2$ .

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Step 11.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 11.4.2

Divide $π$ by $1$ .

$π$

Step 12

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

$x = \frac{π}{12} + π n, \frac{5 π}{12} + π n$ , for any integer $n$

Step 13

Exclude the solutions that do not make $2 \cos (x) + 2 \sin (x) = \sqrt{6}$ true.

$x = \frac{π}{12} + 2 π n, \frac{5 π}{12} + 2 π n$ , for any integer $n$