Trigonometry Examples

$\sin (x + \frac{π}{4}) = \frac{1}{2}$

Step 1

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

$x + \frac{π}{4} = \arcsin (\frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x + \frac{π}{4} = \frac{π}{6}$

Step 3

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

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Step 3.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = \frac{π}{6} - \frac{π}{4}$

Step 3.2

To write $\frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{π}{6} \cdot \frac{2}{2} - \frac{π}{4}$

Step 3.3

To write $- \frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{π}{6} \cdot \frac{2}{2} - \frac{π}{4} \cdot \frac{3}{3}$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{π}{6}$ by $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{6 \cdot 2} - \frac{π}{4} \cdot \frac{3}{3}$

Step 3.4.2

Multiply $6$ by $2$ .

$x = \frac{π \cdot 2}{12} - \frac{π}{4} \cdot \frac{3}{3}$

Step 3.4.3

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$x = \frac{π \cdot 2}{12} - \frac{π \cdot 3}{4 \cdot 3}$

Step 3.4.4

Multiply $4$ by $3$ .

$x = \frac{π \cdot 2}{12} - \frac{π \cdot 3}{12}$

Step 3.5

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π \cdot 3}{12}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π \cdot 3}{12}$

Step 3.6.2

Multiply $3$ by $- 1$ .

$x = \frac{2 π - 3 π}{12}$

Step 3.6.3

Subtract $3 π$ from $2 π$ .

$x = \frac{- π}{12}$

Step 3.7

Move the negative in front of the fraction.

$x = - \frac{π}{12}$

Step 4

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.

$x + \frac{π}{4} = π - \frac{π}{6}$

Step 5

Solve for $x$ .

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Step 5.1

Simplify $π - \frac{π}{6}$ .

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Step 5.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x + \frac{π}{4} = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 5.1.2

Combine fractions.

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Step 5.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$x + \frac{π}{4} = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 5.1.2.2

Combine the numerators over the common denominator.

$x + \frac{π}{4} = \frac{π \cdot 6 - π}{6}$

Step 5.1.3

Simplify the numerator.

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Step 5.1.3.1

Move $6$ to the left of $π$ .

$x + \frac{π}{4} = \frac{6 \cdot π - π}{6}$

Step 5.1.3.2

Subtract $π$ from $6 π$ .

$x + \frac{π}{4} = \frac{5 π}{6}$

Step 5.2

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

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Step 5.2.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = \frac{5 π}{6} - \frac{π}{4}$

Step 5.2.2

To write $\frac{5 π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{5 π}{6} \cdot \frac{2}{2} - \frac{π}{4}$

Step 5.2.3

To write $- \frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{5 π}{6} \cdot \frac{2}{2} - \frac{π}{4} \cdot \frac{3}{3}$

Step 5.2.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.4.1

Multiply $\frac{5 π}{6}$ by $\frac{2}{2}$ .

$x = \frac{5 π \cdot 2}{6 \cdot 2} - \frac{π}{4} \cdot \frac{3}{3}$

Step 5.2.4.2

Multiply $6$ by $2$ .

$x = \frac{5 π \cdot 2}{12} - \frac{π}{4} \cdot \frac{3}{3}$

Step 5.2.4.3

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$x = \frac{5 π \cdot 2}{12} - \frac{π \cdot 3}{4 \cdot 3}$

Step 5.2.4.4

Multiply $4$ by $3$ .

$x = \frac{5 π \cdot 2}{12} - \frac{π \cdot 3}{12}$

Step 5.2.5

Combine the numerators over the common denominator.

$x = \frac{5 π \cdot 2 - π \cdot 3}{12}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Multiply $2$ by $5$ .

$x = \frac{10 π - π \cdot 3}{12}$

Step 5.2.6.2

Multiply $3$ by $- 1$ .

$x = \frac{10 π - 3 π}{12}$

Step 5.2.6.3

Subtract $3 π$ from $10 π$ .

$x = \frac{7 π}{12}$

Step 6

Find the period of $\sin (x + \frac{π}{4})$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

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Step 7.1

Add $2 π$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + 2 π$

Step 7.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$2 π \cdot \frac{12}{12} - \frac{π}{12}$

Step 7.3

Combine fractions.

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Step 7.3.1

Combine $2 π$ and $\frac{12}{12}$ .

$\frac{2 π \cdot 12}{12} - \frac{π}{12}$

Step 7.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 12 - π}{12}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Multiply $12$ by $2$ .

$\frac{24 π - π}{12}$

Step 7.4.2

Subtract $π$ from $24 π$ .

$\frac{23 π}{12}$

Step 7.5

List the new angles.

$x = \frac{23 π}{12}$

Step 8

The period of the $\sin (x + \frac{π}{4})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{7 π}{12} + 2 π n, \frac{23 π}{12} + 2 π n$ , for any integer $n$