Trigonometry Examples

$\sin (π x + (\frac{1}{6} \cdot π)) = 1$

Step 1

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

$π x + \frac{1}{6} \cdot π = \arcsin (1)$

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{1}{6}$ and $π$ .

$π x + \frac{π}{6} = \arcsin (1)$

Step 3

Simplify the right side.

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Step 3.1

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

$π x + \frac{π}{6} = \frac{π}{2}$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$π x = \frac{π}{2} - \frac{π}{6}$

Step 4.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$π x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Step 4.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$π x = \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Step 4.3.2

Multiply $2$ by $3$ .

$π x = \frac{π \cdot 3}{6} - \frac{π}{6}$

Step 4.4

Combine the numerators over the common denominator.

$π x = \frac{π \cdot 3 - π}{6}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Move $3$ to the left of $π$ .

$π x = \frac{3 \cdot π - π}{6}$

Step 4.5.2

Subtract $π$ from $3 π$ .

$π x = \frac{2 π}{6}$

Step 4.6

Cancel the common factor of $2$ and $6$ .

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Step 4.6.1

Factor $2$ out of $2 π$ .

$π x = \frac{2 (π)}{6}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $2$ out of $6$ .

$π x = \frac{2 π}{2 \cdot 3}$

Step 4.6.2.2

Cancel the common factor.

$π x = \frac{2 π}{2 \cdot 3}$

Step 4.6.2.3

Rewrite the expression.

$π x = \frac{π}{3}$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $π$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{π}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{π}$

Step 5.3.2

Cancel the common factor of $π$ .

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Step 5.3.2.1

Cancel the common factor.

$x = \frac{π}{3} \cdot \frac{1}{π}$

Step 5.3.2.2

Rewrite the expression.

$x = \frac{1}{3}$

Step 6

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

$π x + \frac{π}{6} = π - \frac{π}{2}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $π - \frac{π}{2}$ .

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Step 7.1.1

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

$π x + \frac{π}{6} = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 7.1.2

Combine fractions.

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Step 7.1.2.1

Combine $π$ and $\frac{2}{2}$ .

$π x + \frac{π}{6} = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 7.1.2.2

Combine the numerators over the common denominator.

$π x + \frac{π}{6} = \frac{π \cdot 2 - π}{2}$

Step 7.1.3

Simplify the numerator.

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Step 7.1.3.1

Move $2$ to the left of $π$ .

$π x + \frac{π}{6} = \frac{2 \cdot π - π}{2}$

Step 7.1.3.2

Subtract $π$ from $2 π$ .

$π x + \frac{π}{6} = \frac{π}{2}$

Step 7.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.2.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$π x = \frac{π}{2} - \frac{π}{6}$

Step 7.2.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$π x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Step 7.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$π x = \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Step 7.2.3.2

Multiply $2$ by $3$ .

$π x = \frac{π \cdot 3}{6} - \frac{π}{6}$

Step 7.2.4

Combine the numerators over the common denominator.

$π x = \frac{π \cdot 3 - π}{6}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Move $3$ to the left of $π$ .

$π x = \frac{3 \cdot π - π}{6}$

Step 7.2.5.2

Subtract $π$ from $3 π$ .

$π x = \frac{2 π}{6}$

Step 7.2.6

Cancel the common factor of $2$ and $6$ .

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Step 7.2.6.1

Factor $2$ out of $2 π$ .

$π x = \frac{2 (π)}{6}$

Step 7.2.6.2

Cancel the common factors.

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Step 7.2.6.2.1

Factor $2$ out of $6$ .

$π x = \frac{2 π}{2 \cdot 3}$

Step 7.2.6.2.2

Cancel the common factor.

$π x = \frac{2 π}{2 \cdot 3}$

Step 7.2.6.2.3

Rewrite the expression.

$π x = \frac{π}{3}$

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $π$ .

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Step 7.3.2.1.1

Cancel the common factor.

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

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{π}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{π}$

Step 7.3.3.2

Cancel the common factor of $π$ .

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Step 7.3.3.2.1

Cancel the common factor.

$x = \frac{π}{3} \cdot \frac{1}{π}$

Step 7.3.3.2.2

Rewrite the expression.

$x = \frac{1}{3}$

Step 8

Find the period of $\sin (π x + \frac{1}{6} \cdot π)$ .

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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 $π$ in the formula for period.

$\frac{2 π}{| π |}$

Step 8.3

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

$\frac{2 π}{π}$

Step 8.4

Cancel the common factor of $π$ .

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Step 8.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 8.4.2

Divide $2$ by $1$ .

$2$

Step 9

The period of the $\sin (π x + \frac{1}{6} \cdot π)$ function is $2$ so values will repeat every $2$ radians in both directions.

$x = \frac{1}{3} + 2 n$ , for any integer $n$