Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

$x - \frac{π}{2} = \frac{π}{4}$

Step 3

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

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Step 3.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π}{2}$

Step 3.2

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

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

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 $\frac{2}{2}$ .

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

$x = \frac{π + π \cdot 2}{4}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Move $2$ to the left of $π$ .

$x = \frac{π + 2 \cdot π}{4}$

Step 3.5.2

Add $π$ and $2 π$ .

$x = \frac{3 π}{4}$

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.

$x - \frac{π}{2} = π - \frac{π}{4}$

Step 5

Solve for $x$ .

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Step 5.1

Simplify $π - \frac{π}{4}$ .

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Step 5.1.1

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

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

Step 5.1.2

Combine fractions.

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Step 5.1.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 5.1.2.2

Combine the numerators over the common denominator.

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

Step 5.1.3

Simplify the numerator.

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Step 5.1.3.1

Move $4$ to the left of $π$ .

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

Step 5.1.3.2

Subtract $π$ from $4 π$ .

$x - \frac{π}{2} = \frac{3 π}{4}$

Step 5.2

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

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Step 5.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{3 π}{4} + \frac{π}{2}$

Step 5.2.2

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

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

Step 5.2.3

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

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Step 5.2.3.1

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

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

Step 5.2.3.2

Multiply $2$ by $2$ .

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

Step 5.2.4

Combine the numerators over the common denominator.

$x = \frac{3 π + π \cdot 2}{4}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Move $2$ to the left of $π$ .

$x = \frac{3 π + 2 \cdot π}{4}$

Step 5.2.5.2

Add $3 π$ and $2 π$ .

$x = \frac{5 π}{4}$

Step 6

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

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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

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$