Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arcsin (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$2 x + \frac{π}{2} = \frac{π}{4}$

Step 3

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

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Step 3.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$2 x = \frac{π}{4} - \frac{π}{2}$

Step 3.2

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

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

Step 3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

$2 x = \frac{π - π \cdot 2}{4}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $- 1$ .

$2 x = \frac{π - 2 π}{4}$

Step 3.5.2

Subtract $2 π$ from $π$ .

$2 x = \frac{- π}{4}$

Step 3.6

Move the negative in front of the fraction.

$2 x = - \frac{π}{4}$

Step 4

Divide each term in $2 x = - \frac{π}{4}$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = - \frac{π}{4}$ by $2$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{4}}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{4} \cdot \frac{1}{2}$

Step 4.3.2

Multiply $- \frac{π}{4} \cdot \frac{1}{2}$ .

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Step 4.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

$x = - \frac{π}{2 \cdot 4}$

Step 4.3.2.2

Multiply $2$ by $4$ .

$x = - \frac{π}{8}$

Step 5

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.

$2 x + \frac{π}{2} = π - \frac{π}{4}$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $π - \frac{π}{4}$ .

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Step 6.1.1

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

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

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 6.1.2.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Move $4$ to the left of $π$ .

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

Step 6.1.3.2

Subtract $π$ from $4 π$ .

$2 x + \frac{π}{2} = \frac{3 π}{4}$

Step 6.2

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

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Step 6.2.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$2 x = \frac{3 π}{4} - \frac{π}{2}$

Step 6.2.2

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

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

Step 6.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.1

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

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

Step 6.2.3.2

Multiply $2$ by $2$ .

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

Step 6.2.4

Combine the numerators over the common denominator.

$2 x = \frac{3 π - π \cdot 2}{4}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $2$ by $- 1$ .

$2 x = \frac{3 π - 2 π}{4}$

Step 6.2.5.2

Subtract $2 π$ from $3 π$ .

$2 x = \frac{π}{4}$

Step 6.3

Divide each term in $2 x = \frac{π}{4}$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 x = \frac{π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{4}}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{4}}{2}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{2}$

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.

$x = \frac{π}{4} \cdot \frac{1}{2}$

Step 6.3.3.2

Multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 6.3.3.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

$x = \frac{π}{4 \cdot 2}$

Step 6.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

Step 7

Find the period of $\sin (2 x + \frac{π}{2})$ .

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

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Step 8.1

Add $π$ to $- \frac{π}{8}$ to find the positive angle.

$- \frac{π}{8} + π$

Step 8.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$π \cdot \frac{8}{8} - \frac{π}{8}$

Step 8.3

Combine fractions.

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Step 8.3.1

Combine $π$ and $\frac{8}{8}$ .

$\frac{π \cdot 8}{8} - \frac{π}{8}$

Step 8.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 8 - π}{8}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Move $8$ to the left of $π$ .

$\frac{8 \cdot π - π}{8}$

Step 8.4.2

Subtract $π$ from $8 π$ .

$\frac{7 π}{8}$

Step 8.5

List the new angles.

$x = \frac{7 π}{8}$

Step 9

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

$x = \frac{π}{8} + π n, \frac{7 π}{8} + π n$ , for any integer $n$