Trigonometry Examples

$- \sin (π x) = 1$

Step 1

Divide each term in $- \sin (π x) = 1$ by $- 1$ and simplify.

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Step 1.1

Divide each term in $- \sin (π x) = 1$ by $- 1$ .

$\frac{- \sin (π x)}{- 1} = \frac{1}{- 1}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (π x)}{1} = \frac{1}{- 1}$

Step 1.2.2

Divide $\sin (π x)$ by $1$ .

$\sin (π x) = \frac{1}{- 1}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $1$ by $- 1$ .

$\sin (π x) = - 1$

Step 2

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

$π x = \arcsin (- 1)$

Step 3

Simplify the right side.

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Step 3.1

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

$π x = - \frac{π}{2}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $π$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{π}$

Step 4.3.2

Cancel the common factor of $π$ .

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Step 4.3.2.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$x = \frac{- π}{2} \cdot \frac{1}{π}$

Step 4.3.2.2

Factor $π$ out of $- π$ .

$x = \frac{π \cdot - 1}{2} \cdot \frac{1}{π}$

Step 4.3.2.3

Cancel the common factor.

$x = \frac{π \cdot - 1}{2} \cdot \frac{1}{π}$

Step 4.3.2.4

Rewrite the expression.

$x = \frac{- 1}{2}$

Step 4.3.3

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$π x = 2 π + \frac{π}{2} + π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$π x = 2 π + \frac{π}{2} + π - 2 π$

Step 6.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$π x = \frac{3 π}{2}$

Step 6.3

Divide each term in $π x = \frac{3 π}{2}$ by $π$ and simplify.

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Step 6.3.1

Divide each term in $π x = \frac{3 π}{2}$ by $π$ .

$\frac{π x}{π} = \frac{\frac{3 π}{2}}{π}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $π$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{\frac{3 π}{2}}{π}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{π}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{π}$

Step 6.3.3.2

Cancel the common factor of $π$ .

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Step 6.3.3.2.1

Factor $π$ out of $3 π$ .

$x = \frac{π \cdot 3}{2} \cdot \frac{1}{π}$

Step 6.3.3.2.2

Cancel the common factor.

$x = \frac{π \cdot 3}{2} \cdot \frac{1}{π}$

Step 6.3.3.2.3

Rewrite the expression.

$x = \frac{3}{2}$

Step 7

Find the period of $\sin (π x)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| π |}$

Step 7.3

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Step 7.4

Cancel the common factor of $π$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 7.4.2

Divide $2$ by $1$ .

$2$

Step 8

Add $2$ to every negative angle to get positive angles.

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Step 8.1

Add $2$ to $- \frac{1}{2}$ to find the positive angle.

$- \frac{1}{2} + 2$

Step 8.2

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

$2 \cdot \frac{2}{2} - \frac{1}{2}$

Step 8.3

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

$\frac{2 \cdot 2}{2} - \frac{1}{2}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 1}{2}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $2$ by $2$ .

$\frac{4 - 1}{2}$

Step 8.5.2

Subtract $1$ from $4$ .

$\frac{3}{2}$

Step 8.6

List the new angles.

$x = \frac{3}{2}$

Step 9

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

$x = \frac{3}{2} + 2 n, \frac{3}{2} + 2 n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = \frac{3}{2} + 2 n$ , for any integer $n$