Trigonometry Examples

$y = \tan (5 - \sin^{2} (2 x))$

Step 1

Set the argument in $\tan (5 - \sin^{2} (2 x))$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$5 - \sin^{2} (2 x) = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $5$ from both sides of the equation.

$- \sin^{2} (2 x) = \frac{π}{2} + π n - 5$

Step 2.2

Divide each term in $- \sin^{2} (2 x) = \frac{π}{2} + π n - 5$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- \sin^{2} (2 x) = \frac{π}{2} + π n - 5$ by $- 1$ .

$\frac{- \sin^{2} (2 x)}{- 1} = \frac{\frac{π}{2}}{- 1} + \frac{π n}{- 1} + \frac{- 5}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\sin^{2} (2 x)}{1} = \frac{\frac{π}{2}}{- 1} + \frac{π n}{- 1} + \frac{- 5}{- 1}$

Step 2.2.2.2

Divide $\sin^{2} (2 x)$ by $1$ .

$\sin^{2} (2 x) = \frac{\frac{π}{2}}{- 1} + \frac{π n}{- 1} + \frac{- 5}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Move the negative one from the denominator of $\frac{\frac{π}{2}}{- 1}$ .

$\sin^{2} (2 x) = - 1 \cdot \frac{π}{2} + \frac{π n}{- 1} + \frac{- 5}{- 1}$

Step 2.2.3.1.2

Rewrite $- 1 \cdot \frac{π}{2}$ as $- \frac{π}{2}$ .

$\sin^{2} (2 x) = - \frac{π}{2} + \frac{π n}{- 1} + \frac{- 5}{- 1}$

Step 2.2.3.1.3

Move the negative one from the denominator of $\frac{π n}{- 1}$ .

$\sin^{2} (2 x) = - \frac{π}{2} - 1 \cdot (π n) + \frac{- 5}{- 1}$

Step 2.2.3.1.4

Rewrite $- 1 \cdot (π n)$ as $- (π n)$ .

$\sin^{2} (2 x) = - \frac{π}{2} - (π n) + \frac{- 5}{- 1}$

Step 2.2.3.1.5

Divide $- 5$ by $- 1$ .

$\sin^{2} (2 x) = - \frac{π}{2} - π n + 5$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (2 x) = \pm \sqrt{- \frac{π}{2} - π n + 5}$

Step 2.4

Simplify $\pm \sqrt{- \frac{π}{2} - π n + 5}$ .

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Step 2.4.1

Reorder terms.

$\sin (2 x) = \pm \sqrt{- π n + 5 - \frac{π}{2}}$

Step 2.4.2

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

$\sin (2 x) = \pm \sqrt{- π n \cdot \frac{2}{2} - \frac{π}{2} + 5}$

Step 2.4.3

Combine $- π n$ and $\frac{2}{2}$ .

$\sin (2 x) = \pm \sqrt{\frac{- π n \cdot 2}{2} - \frac{π}{2} + 5}$

Step 2.4.4

Combine the numerators over the common denominator.

$\sin (2 x) = \pm \sqrt{\frac{- π n \cdot 2 - π}{2} + 5}$

Step 2.4.5

Multiply $2$ by $- 1$ .

$\sin (2 x) = \pm \sqrt{\frac{- 2 π n - π}{2} + 5}$

Step 2.4.6

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

$\sin (2 x) = \pm \sqrt{\frac{- 2 π n - π}{2} + 5 \cdot \frac{2}{2}}$

Step 2.4.7

Simplify terms.

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Step 2.4.7.1

Combine $5$ and $\frac{2}{2}$ .

$\sin (2 x) = \pm \sqrt{\frac{- 2 π n - π}{2} + \frac{5 \cdot 2}{2}}$

Step 2.4.7.2

Combine the numerators over the common denominator.

$\sin (2 x) = \pm \sqrt{\frac{- 2 π n - π + 5 \cdot 2}{2}}$

Step 2.4.8

Simplify the numerator.

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Step 2.4.8.1

Multiply $5$ by $2$ .

$\sin (2 x) = \pm \sqrt{\frac{- 2 π n - π + 10}{2}}$

Step 2.4.8.2

Reorder terms.

$\sin (2 x) = \pm \sqrt{\frac{- 2 π n + 10 - π}{2}}$

Step 2.4.9

Rewrite $\sqrt{\frac{- 2 π n + 10 - π}{2}}$ as $\frac{\sqrt{- 2 π n + 10 - π}}{\sqrt{2}}$ .

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π}}{\sqrt{2}}$

Step 2.4.10

Multiply $\frac{\sqrt{- 2 π n + 10 - π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.4.11

Combine and simplify the denominator.

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Step 2.4.11.1

Multiply $\frac{\sqrt{- 2 π n + 10 - π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.4.11.2

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

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.11.3

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

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.11.4

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

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.11.5

Add $1$ and $1$ .

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.4.11.6

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

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Step 2.4.11.6.1

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

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.4.11.6.2

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

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.4.11.6.3

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

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.11.6.4

Cancel the common factor of $2$ .

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Step 2.4.11.6.4.1

Cancel the common factor.

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.11.6.4.2

Rewrite the expression.

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{2^{1}}$

Step 2.4.11.6.5

Evaluate the exponent.

$\sin (2 x) = \pm \frac{\sqrt{- 2 π n + 10 - π} \sqrt{2}}{2}$

Step 2.4.12

Combine using the product rule for radicals.

$\sin (2 x) = \pm \frac{\sqrt{(- 2 π n + 10 - π) \cdot 2}}{2}$

Step 2.4.13

Reorder factors in $\pm \frac{\sqrt{(- 2 π n + 10 - π) \cdot 2}}{2}$ .

$\sin (2 x) = \pm \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$\sin (2 x) = \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (2 x) = - \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (2 x) = \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}, - \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$

Step 2.6

Set up each of the solutions to solve for $x$ .

$\sin (2 x) = \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$

$\sin (2 x) = - \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$

Step 2.7

Solve for $x$ in $\sin (2 x) = \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$ .

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Step 2.7.1

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

$2 x = \arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})$

Step 2.7.2

Divide each term in $2 x = \arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})$ by $2$ and simplify.

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Step 2.7.2.1

Divide each term in $2 x = \arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$

Step 2.7.2.2

Simplify the left side.

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Step 2.7.2.2.1

Cancel the common factor of $2$ .

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Step 2.7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$

Step 2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$

Step 2.8

Solve for $x$ in $\sin (2 x) = - \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2}$ .

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Step 2.8.1

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

$2 x = \arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})$

Step 2.8.2

Divide each term in $2 x = \arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})$ by $2$ and simplify.

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Step 2.8.2.1

Divide each term in $2 x = \arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $2$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$

Step 2.9

List all of the solutions.

$x = \frac{\arcsin (\frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}, \frac{\arcsin (- \frac{\sqrt{2 (- 2 π n + 10 - π)}}{2})}{2}$ , for any integer $n$

Step 2.10

Consolidate the answers.

$x = - 0.28301993 + 0.28301993 n$ , for any integer $n$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = - 0.28301993 + 0.28301993 n}$ $n$ , for any integer $n$

Step 4