Trigonometry Examples

$\sin (2 x - \frac{π}{2}) = - 1$

Step 1

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

$2 x - \frac{π}{2} = \arcsin (- 1)$

Step 2

Simplify the right side.

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Step 2.1

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

$2 x - \frac{π}{2} = - \frac{π}{2}$

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.

$2 x = - \frac{π}{2} + \frac{π}{2}$

Step 3.2

Combine the numerators over the common denominator.

$2 x = \frac{- π + π}{2}$

Step 3.3

Add $- π$ and $π$ .

$2 x = \frac{0}{2}$

Step 3.4

Divide $0$ by $2$ .

$2 x = 0$

Step 4

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

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.

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$x = 0$

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.

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

Step 6

Simplify the expression to find the second solution.

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Step 6.1

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

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

Step 6.2

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

$2 x - \frac{π}{2} = \frac{3 π}{2}$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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Step 6.3.1.1

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

$2 x = \frac{3 π}{2} + \frac{π}{2}$

Step 6.3.1.2

Combine the numerators over the common denominator.

$2 x = \frac{3 π + π}{2}$

Step 6.3.1.3

Add $3 π$ and $π$ .

$2 x = \frac{4 π}{2}$

Step 6.3.1.4

Cancel the common factor of $4$ and $2$ .

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Step 6.3.1.4.1

Factor $2$ out of $4 π$ .

$2 x = \frac{2 (2 π)}{2}$

Step 6.3.1.4.2

Cancel the common factors.

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Step 6.3.1.4.2.1

Factor $2$ out of $2$ .

$2 x = \frac{2 (2 π)}{2 (1)}$

Step 6.3.1.4.2.2

Cancel the common factor.

$2 x = \frac{2 (2 π)}{2 \cdot 1}$

Step 6.3.1.4.2.3

Rewrite the expression.

$2 x = \frac{2 π}{1}$

Step 6.3.1.4.2.4

Divide $2 π$ by $1$ .

$2 x = 2 π$

Step 6.3.2

Divide each term in $2 x = 2 π$ by $2$ and simplify.

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Step 6.3.2.1

Divide each term in $2 x = 2 π$ by $2$ .

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Cancel the common factor of $2$ .

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Step 6.3.2.3.1.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 6.3.2.3.1.2

Divide $π$ by $1$ .

$x = π$

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

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

$x = π n, π + π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = π n$ , for any integer $n$