Finite Math Examples

\frac{4 sin (A) \cdot cos (A) \cdot cos (2 A) \cdot sin (15)}{sin (2 A) (tan (225) - 2 {sin}^{2} (A))}

$\frac{4 \sin (A) \cdot \cos (A) \cdot \cos (2 A) \cdot \sin (15)}{\sin (2 A) (\tan (225) - 2 \sin^{2} (A))}$

Step 1

Set the denominator in

$\frac{4 \sin (A) \cdot \cos (A) \cdot \cos (2 A) \cdot \sin (15)}{\sin (2 A) (\tan (225) - 2 \sin^{2} (A))}$ equal to

$0$ to find where the expression is undefined.

$\sin (2 A) (\tan (225) - 2 \sin^{2} (A)) = 0$

Step 2

Solve for

$A$ .

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Step 2.1

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$\sin (2 A) = 0$

$\tan (225) - 2 \sin^{2} (A) = 0$

Step 2.2

Set

$\sin (2 A)$ equal to

$0$ and solve for

$A$ .

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Step 2.2.1

Set

$\sin (2 A)$ equal to

$0$ .

$\sin (2 A) = 0$

Step 2.2.2

Solve

$\sin (2 A) = 0$ for

$A$ .

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Step 2.2.2.1

Take the inverse sine of both sides of the equation to extract

$A$ from inside the sine.

$2 A = \arcsin (0)$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

The exact value of

$\arcsin (0)$ is

$0$ .

$2 A = 0$

Step 2.2.2.3

Divide each term in

$2 A = 0$ by

$2$ and simplify.

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Step 2.2.2.3.1

Divide each term in

$2 A = 0$ by

$2$ .

$\frac{2 A}{2} = \frac{0}{2}$

Step 2.2.2.3.2

Simplify the left side.

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Step 2.2.2.3.2.1

Cancel the common factor of

$2$ .

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Step 2.2.2.3.2.1.1

Cancel the common factor.

$\frac{2 A}{2} = \frac{0}{2}$

Step 2.2.2.3.2.1.2

Divide

$A$ by

$1$ .

$A = \frac{0}{2}$

Step 2.2.2.3.3

Simplify the right side.

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Step 2.2.2.3.3.1

Divide

$0$ by

$2$ .

$A = 0$

Step 2.2.2.4

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

$180$ to find the solution in the second quadrant.

$2 A = 180 - 0$

Step 2.2.2.5

Solve for

$A$ .

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Step 2.2.2.5.1

Simplify.

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Step 2.2.2.5.1.1

Multiply

$- 1$ by

$0$ .

$2 A = 180 + 0$

Step 2.2.2.5.1.2

Add

$180$ and

$0$ .

$2 A = 180$

Step 2.2.2.5.2

Divide each term in

$2 A = 180$ by

$2$ and simplify.

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Step 2.2.2.5.2.1

Divide each term in

$2 A = 180$ by

$2$ .

$\frac{2 A}{2} = \frac{180}{2}$

Step 2.2.2.5.2.2

Simplify the left side.

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Step 2.2.2.5.2.2.1

Cancel the common factor of

$2$ .

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Step 2.2.2.5.2.2.1.1

Cancel the common factor.

$\frac{2 A}{2} = \frac{180}{2}$

Step 2.2.2.5.2.2.1.2

Divide

$A$ by

$1$ .

$A = \frac{180}{2}$

Step 2.2.2.5.2.3

Simplify the right side.

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Step 2.2.2.5.2.3.1

Divide

$180$ by

$2$ .

$A = 90$

Step 2.2.2.6

Find the period of

$\sin (2 A)$ .

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Step 2.2.2.6.1

The period of the function can be calculated using

$\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 2.2.2.6.2

Replace

$b$ with

$2$ in the formula for period.

$\frac{360}{| 2 |}$

Step 2.2.2.6.3

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

$0$ and

$2$ is

$2$ .

$\frac{360}{2}$

Step 2.2.2.6.4

Divide

$360$ by

$2$ .

$180$

Step 2.2.2.7

The period of the

$\sin (2 A)$ function is

$180$ so values will repeat every

$180$ degrees in both directions.

$A = 180 n, 90 + 180 n$ , for any integer

$n$

$A = 180 n, 90 + 180 n$ , for any integer

$n$

$A = 180 n, 90 + 180 n$ , for any integer

$n$

Step 2.3

Set

$\tan (225) - 2 \sin^{2} (A)$ equal to

$0$ and solve for

$A$ .

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Step 2.3.1

Set

$\tan (225) - 2 \sin^{2} (A)$ equal to

$0$ .

$\tan (225) - 2 \sin^{2} (A) = 0$

Step 2.3.2

Solve

$\tan (225) - 2 \sin^{2} (A) = 0$ for

$A$ .

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Simplify each term.

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Step 2.3.2.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\tan (45) - 2 \sin^{2} (A) = 0$

Step 2.3.2.1.1.2

The exact value of

$\tan (45)$ is

$1$ .

$1 - 2 \sin^{2} (A) = 0$

Step 2.3.2.2

Subtract

$1$ from both sides of the equation.

$- 2 \sin^{2} (A) = - 1$

Step 2.3.2.3

Divide each term in

$- 2 \sin^{2} (A) = - 1$ by

$- 2$ and simplify.

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Step 2.3.2.3.1

Divide each term in

$- 2 \sin^{2} (A) = - 1$ by

$- 2$ .

$\frac{- 2 \sin^{2} (A)}{- 2} = \frac{- 1}{- 2}$

Step 2.3.2.3.2

Simplify the left side.

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Step 2.3.2.3.2.1

Cancel the common factor of

$- 2$ .

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Step 2.3.2.3.2.1.1

Cancel the common factor.

$\frac{- 2 \sin^{2} (A)}{- 2} = \frac{- 1}{- 2}$

Step 2.3.2.3.2.1.2

Divide

$\sin^{2} (A)$ by

$1$ .

$\sin^{2} (A) = \frac{- 1}{- 2}$

Step 2.3.2.3.3

Simplify the right side.

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Step 2.3.2.3.3.1

Dividing two negative values results in a positive value.

$\sin^{2} (A) = \frac{1}{2}$

Step 2.3.2.4

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

$\sin (A) = \pm \sqrt{\frac{1}{2}}$

Step 2.3.2.5

Simplify

$\pm \sqrt{\frac{1}{2}}$ .

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Step 2.3.2.5.1

Rewrite

$\sqrt{\frac{1}{2}}$ as

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

$\sin (A) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.3.2.5.2

Any root of

$1$ is

$1$ .

$\sin (A) = \pm \frac{1}{\sqrt{2}}$

Step 2.3.2.5.3

Multiply

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

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

$\sin (A) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.3.2.5.4

Combine and simplify the denominator.

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Step 2.3.2.5.4.1

Multiply

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

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

$\sin (A) = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.3.2.5.4.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sin (A) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.3.2.5.4.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sin (A) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.3.2.5.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (A) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.3.2.5.4.5

Add

$1$ and

$1$ .

$\sin (A) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.3.2.5.4.6

Rewrite

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

$2$ .

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Step 2.3.2.5.4.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\sin (A) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.3.2.5.4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sin (A) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.5.4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sin (A) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.3.2.5.4.6.4

Cancel the common factor of

$2$ .

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Step 2.3.2.5.4.6.4.1

Cancel the common factor.

$\sin (A) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.3.2.5.4.6.4.2

Rewrite the expression.

$\sin (A) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 2.3.2.5.4.6.5

Evaluate the exponent.

$\sin (A) = \pm \frac{\sqrt{2}}{2}$

Step 2.3.2.6

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

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Step 2.3.2.6.1

First, use the positive value of the

$\pm$ to find the first solution.

$\sin (A) = \frac{\sqrt{2}}{2}$

Step 2.3.2.6.2

Next, use the negative value of the

$\pm$ to find the second solution.

$\sin (A) = - \frac{\sqrt{2}}{2}$

Step 2.3.2.6.3

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

$\sin (A) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 2.3.2.7

Set up each of the solutions to solve for

$A$ .

$\sin (A) = \frac{\sqrt{2}}{2}$

$\sin (A) = - \frac{\sqrt{2}}{2}$

Step 2.3.2.8

Solve for

$A$ in

$\sin (A) = \frac{\sqrt{2}}{2}$ .

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Step 2.3.2.8.1

Take the inverse sine of both sides of the equation to extract

$A$ from inside the sine.

$A = \arcsin (\frac{\sqrt{2}}{2})$

Step 2.3.2.8.2

Simplify the right side.

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Step 2.3.2.8.2.1

The exact value of

$\arcsin (\frac{\sqrt{2}}{2})$ is

$45$ .

$A = 45$

Step 2.3.2.8.3

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

$180$ to find the solution in the second quadrant.

$A = 180 - 45$

Step 2.3.2.8.4

Subtract

$45$ from

$180$ .

$A = 135$

Step 2.3.2.8.5

Find the period of

$\sin (A)$ .

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Step 2.3.2.8.5.1

The period of the function can be calculated using

$\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 2.3.2.8.5.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 2.3.2.8.5.3

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

$0$ and

$1$ is

$1$ .

$\frac{360}{1}$

Step 2.3.2.8.5.4

Divide

$360$ by

$1$ .

$360$

Step 2.3.2.8.6

The period of the

$\sin (A)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$A = 45 + 360 n, 135 + 360 n$ , for any integer

$n$

$A = 45 + 360 n, 135 + 360 n$ , for any integer

$n$

Step 2.3.2.9

Solve for

$A$ in

$\sin (A) = - \frac{\sqrt{2}}{2}$ .

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Step 2.3.2.9.1

Take the inverse sine of both sides of the equation to extract

$A$ from inside the sine.

$A = \arcsin (- \frac{\sqrt{2}}{2})$

Step 2.3.2.9.2

Simplify the right side.

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Step 2.3.2.9.2.1

The exact value of

$\arcsin (- \frac{\sqrt{2}}{2})$ is

$- 45$ .

$A = - 45$

Step 2.3.2.9.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

$180$ to find the solution in the third quadrant.

$A = 360 + 45 + 180$

Step 2.3.2.9.4

Simplify the expression to find the second solution.

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Step 2.3.2.9.4.1

Subtract

$360 °$ from

$360 + 45 + 180 °$ .

$A = 360 + 45 + 180 ° - 360 °$

Step 2.3.2.9.4.2

The resulting angle of

$225 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 45 + 180$ .

$A = 225 °$

Step 2.3.2.9.5

Find the period of

$\sin (A)$ .

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Step 2.3.2.9.5.1

The period of the function can be calculated using

$\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 2.3.2.9.5.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 2.3.2.9.5.3

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

$0$ and

$1$ is

$1$ .

$\frac{360}{1}$

Step 2.3.2.9.5.4

Divide

$360$ by

$1$ .

$360$

Step 2.3.2.9.6

Add

$360$ to every negative angle to get positive angles.

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Step 2.3.2.9.6.1

Add

$360$ to

$- 45$ to find the positive angle.

$- 45 + 360$

Step 2.3.2.9.6.2

Subtract

$45$ from

$360$ .

$315$

Step 2.3.2.9.6.3

List the new angles.

$A = 315$

Step 2.3.2.9.7

The period of the

$\sin (A)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$A = 225 + 360 n, 315 + 360 n$ , for any integer

$n$

$A = 225 + 360 n, 315 + 360 n$ , for any integer

$n$

Step 2.3.2.10

List all of the solutions.

$A = 45 + 360 n, 135 + 360 n, 225 + 360 n, 315 + 360 n$ , for any integer

$n$

Step 2.3.2.11

Consolidate the answers.

$A = 45 + 90 n$ , for any integer

$n$

$A = 45 + 90 n$ , for any integer

$n$

$A = 45 + 90 n$ , for any integer

$n$

Step 2.4

The final solution is all the values that make

$\sin (2 A) (\tan (225) - 2 \sin^{2} (A)) = 0$ true.

$A = 180 n, 90 + 180 n, 45 + 90 n$ , for any integer

$n$

Step 2.5

Consolidate the answers.

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Step 2.5.1

Consolidate

$180 n$ and

$90 + 180 n$ to

$90 n$ .

$A = 90 n, 45 + 90 n$ , for any integer

$n$

Step 2.5.2

Consolidate the answers.

$A = 45 n$ , for any integer

$n$

$A = 45 n$ , for any integer

$n$

$A = 45 n$ , for any integer

$n$

Step 3

The domain is all values of

$A$ that make the expression defined.

Set-Builder Notation:

${A | A \neq 45 n}$ , for any integer

$n$

Step 4