Trigonometry Examples

$\sqrt{2} \tan (x) = 2 \sin (x)$

Step 1

Divide each term in $\sqrt{2} \tan (x) = 2 \sin (x)$ by $\sqrt{2} \tan (x)$ and simplify.

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Step 1.1

Divide each term in $\sqrt{2} \tan (x) = 2 \sin (x)$ by $\sqrt{2} \tan (x)$ .

$\frac{\sqrt{2} \tan (x)}{\sqrt{2} \tan (x)} = \frac{2 \sin (x)}{\sqrt{2} \tan (x)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\sqrt{2}$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} \tan (x)}{\sqrt{2} \tan (x)} = \frac{2 \sin (x)}{\sqrt{2} \tan (x)}$

Step 1.2.1.2

Rewrite the expression.

$\frac{\tan (x)}{\tan (x)} = \frac{2 \sin (x)}{\sqrt{2} \tan (x)}$

Step 1.2.2

Cancel the common factor of $\tan (x)$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{\tan (x)}{\tan (x)} = \frac{2 \sin (x)}{\sqrt{2} \tan (x)}$

Step 1.2.2.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Separate fractions.

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

Step 1.3.2

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

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

Step 1.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.3.4

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.3.5

Cancel the common factor of $\sin (x)$ .

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Step 1.3.5.1

Cancel the common factor.

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

Step 1.3.5.2

Rewrite the expression.

$1 = \frac{2}{\sqrt{2}} \cos (x)$

Step 1.3.6

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.3.7

Combine and simplify the denominator.

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Step 1.3.7.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$1 = \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} \cos (x)$

Step 1.3.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$1 = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \cos (x)$

Step 1.3.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$1 = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \cos (x)$

Step 1.3.7.4

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

$1 = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cos (x)$

Step 1.3.7.5

Add $1$ and $1$ .

$1 = \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} \cos (x)$

Step 1.3.7.6

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

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Step 1.3.7.6.1

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

$1 = \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cos (x)$

Step 1.3.7.6.2

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

$1 = \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cos (x)$

Step 1.3.7.6.3

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

$1 = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \cos (x)$

Step 1.3.7.6.4

Cancel the common factor of $2$ .

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Step 1.3.7.6.4.1

Cancel the common factor.

$1 = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \cos (x)$

Step 1.3.7.6.4.2

Rewrite the expression.

$1 = \frac{2 \sqrt{2}}{2^{1}} \cos (x)$

Step 1.3.7.6.5

Evaluate the exponent.

$1 = \frac{2 \sqrt{2}}{2} \cos (x)$

Step 1.3.8

Cancel the common factor of $2$ .

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Step 1.3.8.1

Cancel the common factor.

$1 = \frac{2 \sqrt{2}}{2} \cos (x)$

Step 1.3.8.2

Divide $\sqrt{2}$ by $1$ .

$1 = \sqrt{2} \cos (x)$

Step 2

Rewrite the equation as $\sqrt{2} \cos (x) = 1$ .

$\sqrt{2} \cos (x) = 1$

Step 3

Divide each term in $\sqrt{2} \cos (x) = 1$ by $\sqrt{2}$ and simplify.

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Step 3.1

Divide each term in $\sqrt{2} \cos (x) = 1$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \cos (x)}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\sqrt{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} \cos (x)}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

Step 3.2.1.2

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

$\cos (x) = \frac{1}{\sqrt{2}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.3.2

Combine and simplify the denominator.

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Step 3.3.2.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.3.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.3.2.4

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

$\cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.3.2.5

Add $1$ and $1$ .

$\cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.3.2.6

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

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Step 3.3.2.6.1

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

$\cos (x) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.3.2.6.2

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

$\cos (x) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.3.2.6.3

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

$\cos (x) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.3.2.6.4

Cancel the common factor of $2$ .

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Step 3.3.2.6.4.1

Cancel the common factor.

$\cos (x) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.3.2.6.4.2

Rewrite the expression.

$\cos (x) = \frac{\sqrt{2}}{2^{1}}$

Step 3.3.2.6.5

Evaluate the exponent.

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 4

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

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 6

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.

$x = 2 π - \frac{π}{4}$

Step 7

Simplify $2 π - \frac{π}{4}$ .

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Step 7.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 7.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 8

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

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 1 |}$

Step 8.3

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

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$