Trigonometry Examples

$\sec (arccot (\frac{\sqrt{64 - u^{2}}}{u}))$

Step 1

Set the denominator in $\frac{\sqrt{64 - u^{2}}}{u}$ equal to $0$ to find where the expression is undefined.

$u = 0$

Step 2

Set the radicand in $\sqrt{64 - u^{2}}$ less than $0$ to find where the expression is undefined.

$64 - u^{2} < 0$

Step 3

Solve for $u$ .

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Step 3.1

Subtract $64$ from both sides of the inequality.

$- u^{2} < - 64$

Step 3.2

Divide each term in $- u^{2} < - 64$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- u^{2} < - 64$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- u^{2}}{- 1} > \frac{- 64}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{u^{2}}{1} > \frac{- 64}{- 1}$

Step 3.2.2.2

Divide $u^{2}$ by $1$ .

$u^{2} > \frac{- 64}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 64$ by $- 1$ .

$u^{2} > 64$

Step 3.3

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

$\sqrt{u^{2}} > \sqrt{64}$

Step 3.4

Simplify the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Pull terms out from under the radical.

$| u | > \sqrt{64}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\sqrt{64}$ .

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Step 3.4.2.1.1

Rewrite $64$ as $8^{2}$ .

$| u | > \sqrt{8^{2}}$

Step 3.4.2.1.2

Pull terms out from under the radical.

$| u | > | 8 |$

Step 3.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$| u | > 8$

Step 3.5

Write $| u | > 8$ as a piecewise.

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Step 3.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$u \geq 0$

Step 3.5.2

In the piece where $u$ is non-negative, remove the absolute value.

$u > 8$

Step 3.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$u < 0$

Step 3.5.4

In the piece where $u$ is negative, remove the absolute value and multiply by $- 1$ .

$- u > 8$

Step 3.5.5

Write as a piecewise.

${\begin{cases} u > 8 & u \geq 0 \\ - u > 8 & u < 0 \end{cases}$

Step 3.6

Find the intersection of $u > 8$ and $u \geq 0$ .

$u > 8$

Step 3.7

Divide each term in $- u > 8$ by $- 1$ and simplify.

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Step 3.7.1

Divide each term in $- u > 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- u}{- 1} < \frac{8}{- 1}$

Step 3.7.2

Simplify the left side.

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Step 3.7.2.1

Dividing two negative values results in a positive value.

$\frac{u}{1} < \frac{8}{- 1}$

Step 3.7.2.2

Divide $u$ by $1$ .

$u < \frac{8}{- 1}$

Step 3.7.3

Simplify the right side.

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Step 3.7.3.1

Divide $8$ by $- 1$ .

$u < - 8$

Step 3.8

Find the union of the solutions.

$u < - 8$ or $u > 8$

Step 4

Set the argument in $\sec (arccot (\frac{\sqrt{64 - u^{2}}}{u}))$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$arccot (\frac{\sqrt{64 - u^{2}}}{u}) = \frac{π}{2} + π n$ , for any integer $n$

Step 5

Solve for $u$ .

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Step 5.1

Take the inverse arccotangent of both sides of the equation to extract $u$ from inside the arccotangent.

$\frac{\sqrt{64 - u^{2}}}{u} = \cot (\frac{π}{2} + π n)$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{8^{2} - u^{2}}}{u} = \cot (\frac{π}{2} + π n)$

Step 5.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = u$ .

$\frac{\sqrt{(8 + u) (8 - u)}}{u} = \cot (\frac{π}{2} + π n)$

Step 5.3

Cross multiply.

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Step 5.3.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$\cot (\frac{π}{2} + π n) \cdot (u) = \sqrt{(8 + u) (8 - u)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Reorder factors in $\cot (\frac{π}{2} + π n) u$ .

$u \cot (\frac{π}{2} + π n) = \sqrt{(8 + u) (8 - u)}$

Step 5.4

Rewrite the equation as $\sqrt{(8 + u) (8 - u)} = u \cot (\frac{π}{2} + π n)$ .

$\sqrt{(8 + u) (8 - u)} = u \cot (\frac{π}{2} + π n)$

Step 5.5

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{(8 + u) (8 - u)}}^{2} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6

Simplify each side of the equation.

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(8 + u) (8 - u)}$ as ${((8 + u) (8 - u))}^{\frac{1}{2}}$ .

${({((8 + u) (8 - u))}^{\frac{1}{2}})}^{2} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Simplify ${({((8 + u) (8 - u))}^{\frac{1}{2}})}^{2}$ .

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Step 5.6.2.1.1

Multiply the exponents in ${({((8 + u) (8 - u))}^{\frac{1}{2}})}^{2}$ .

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Step 5.6.2.1.1.1

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

${((8 + u) (8 - u))}^{\frac{1}{2} \cdot 2} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.6.2.1.1.2.1

Cancel the common factor.

${((8 + u) (8 - u))}^{\frac{1}{2} \cdot 2} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.1.2.2

Rewrite the expression.

${((8 + u) (8 - u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.2

Expand $(8 + u) (8 - u)$ using the FOIL Method.

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Step 5.6.2.1.2.1

Apply the distributive property.

${(8 (8 - u) + u (8 - u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.2.2

Apply the distributive property.

${(8 \cdot 8 + 8 (- u) + u (8 - u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.2.3

Apply the distributive property.

${(8 \cdot 8 + 8 (- u) + u \cdot 8 + u (- u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3

Simplify and combine like terms.

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Step 5.6.2.1.3.1

Simplify each term.

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Step 5.6.2.1.3.1.1

Multiply $8$ by $8$ .

${(64 + 8 (- u) + u \cdot 8 + u (- u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.1.2

Multiply $- 1$ by $8$ .

${(64 - 8 u + u \cdot 8 + u (- u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.1.3

Move $8$ to the left of $u$ .

${(64 - 8 u + 8 \cdot u + u (- u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(64 - 8 u + 8 u - u \cdot u)}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.1.5

Multiply $u$ by $u$ by adding the exponents.

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Step 5.6.2.1.3.1.5.1

Move $u$ .

${(64 - 8 u + 8 u - (u \cdot u))}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.1.5.2

Multiply $u$ by $u$ .

${(64 - 8 u + 8 u - u^{2})}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.2

Add $- 8 u$ and $8 u$ .

${(64 + 0 - u^{2})}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.3.3

Add $64$ and $0$ .

${(64 - u^{2})}^{1} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.2.1.4

Simplify.

$64 - u^{2} = {(u \cot (\frac{π}{2} + π n))}^{2}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Apply the product rule to $u \cot (\frac{π}{2} + π n)$ .

$64 - u^{2} = u^{2} \cot^{2} (\frac{π}{2} + π n)$

Step 5.7

Solve for $u$ .

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Step 5.7.1

Move all terms containing $u$ to the left side of the equation.

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Step 5.7.1.1

Subtract $u^{2} \cot^{2} (\frac{π}{2} + π n)$ from both sides of the equation.

$64 - u^{2} - u^{2} \cot^{2} (\frac{π}{2} + π n) = 0$

Step 5.7.1.2

Factor $- u^{2}$ out of $- u^{2}$ .

$64 - u^{2} (1) - u^{2} \cot^{2} (\frac{π}{2} + π n) = 0$

Step 5.7.1.3

Factor $- u^{2}$ out of $- u^{2} \cot^{2} (\frac{π}{2} + π n)$ .

$64 - u^{2} (1) - u^{2} (\cot^{2} (\frac{π}{2} + π n)) = 0$

Step 5.7.1.4

Factor $- u^{2}$ out of $- u^{2} (1) - u^{2} (\cot^{2} (\frac{π}{2} + π n))$ .

$64 - u^{2} (1 + \cot^{2} (\frac{π}{2} + π n)) = 0$

Step 5.7.1.5

Apply pythagorean identity.

$64 - u^{2} \csc^{2} (\frac{π}{2} + π n) = 0$

Step 5.7.2

Subtract $64$ from both sides of the equation.

$- u^{2} \csc^{2} (\frac{π}{2} + π n) = - 64$

Step 5.7.3

Divide each term in $- u^{2} \csc^{2} (\frac{π}{2} + π n) = - 64$ by $- \csc^{2} (\frac{π}{2} + π n)$ and simplify.

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Step 5.7.3.1

Divide each term in $- u^{2} \csc^{2} (\frac{π}{2} + π n) = - 64$ by $- \csc^{2} (\frac{π}{2} + π n)$ .

$\frac{- u^{2} \csc^{2} (\frac{π}{2} + π n)}{- \csc^{2} (\frac{π}{2} + π n)} = \frac{- 64}{- \csc^{2} (\frac{π}{2} + π n)}$

Step 5.7.3.2

Simplify the left side.

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Step 5.7.3.2.1

Dividing two negative values results in a positive value.

$\frac{u^{2} \csc^{2} (\frac{π}{2} + π n)}{\csc^{2} (\frac{π}{2} + π n)} = \frac{- 64}{- \csc^{2} (\frac{π}{2} + π n)}$

Step 5.7.3.2.2

Cancel the common factor of $\csc^{2} (\frac{π}{2} + π n)$ .

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Step 5.7.3.2.2.1

Cancel the common factor.

$\frac{u^{2} \csc^{2} (\frac{π}{2} + π n)}{\csc^{2} (\frac{π}{2} + π n)} = \frac{- 64}{- \csc^{2} (\frac{π}{2} + π n)}$

Step 5.7.3.2.2.2

Divide $u^{2}$ by $1$ .

$u^{2} = \frac{- 64}{- \csc^{2} (\frac{π}{2} + π n)}$

Step 5.7.3.3

Simplify the right side.

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Step 5.7.3.3.1

Dividing two negative values results in a positive value.

$u^{2} = \frac{64}{\csc^{2} (\frac{π}{2} + π n)}$

Step 5.7.4

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

$u = \pm \sqrt{\frac{64}{\csc^{2} (\frac{π}{2} + π n)}}$

Step 5.7.5

Simplify $\pm \sqrt{\frac{64}{\csc^{2} (\frac{π}{2} + π n)}}$ .

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Step 5.7.5.1

Rewrite $64$ as $8^{2}$ .

$u = \pm \sqrt{\frac{8^{2}}{\csc^{2} (\frac{π}{2} + π n)}}$

Step 5.7.5.2

Rewrite $\frac{8^{2}}{\csc^{2} (\frac{π}{2} + π n)}$ as ${(\frac{8}{\csc (\frac{π}{2} + π n)})}^{2}$ .

$u = \pm \sqrt{{(\frac{8}{\csc (\frac{π}{2} + π n)})}^{2}}$

Step 5.7.5.3

Pull terms out from under the radical, assuming positive real numbers.

$u = \pm \frac{8}{\csc (\frac{π}{2} + π n)}$

Step 5.7.5.4

Separate fractions.

$u = \pm \frac{8}{1} \cdot \frac{1}{\csc (\frac{π}{2} + π n)}$

Step 5.7.5.5

Rewrite $\csc (\frac{π}{2} + π n)$ in terms of sines and cosines.

$u = \pm \frac{8}{1} \cdot \frac{1}{\frac{1}{\sin (\frac{π}{2} + π n)}}$

Step 5.7.5.6

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (\frac{π}{2} + π n)}$ .

$u = \pm \frac{8}{1} (1 \sin (\frac{π}{2} + π n))$

Step 5.7.5.7

Multiply $\sin (\frac{π}{2} + π n)$ by $1$ .

$u = \pm \frac{8}{1} \sin (\frac{π}{2} + π n)$

Step 5.7.5.8

Divide $8$ by $1$ .

$u = \pm 8 \sin (\frac{π}{2} + π n)$

Step 5.7.6

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

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Step 5.7.6.1

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

$u = 8 \sin (\frac{π}{2} + π n)$

Step 5.7.6.2

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