Trigonometry Examples

$\sin (x) \cos (x) = 0$

Multiply each term in $\sin (x) \cos (x) = 0$ by $2$ to eliminate the fractions.

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Multiply each term in $\sin (x) \cos (x) = 0$ by $2$ .

$\sin (x) \cos (x) \cdot 2 = 0 \cdot 2$

Simplify the left side.

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Reorder $\sin (x) \cos (x)$ and $2$ .

$2 \cdot (\sin (x) \cos (x)) = 0 \cdot 2$

Apply the sine double-angle identity.

$\sin (2 x) = 0 \cdot 2$

Simplify the right side.

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Multiply $0$ by $2$ .

$\sin (2 x) = 0$

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

$2 x = \arcsin (0)$

Simplify the right side.

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The exact value of $\arcsin (0)$ is $0$ .

$2 x = 0$

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

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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Divide $0$ by $2$ .

$x = 0$

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.

$2 x = π - 0$

Solve for $x$ .

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Simplify.

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Multiply $- 1$ by $0$ .

$2 x = π + 0$

Add $π$ and $0$ .

$2 x = π$

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

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

$\frac{2 x}{2} = \frac{π}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Find the period of $\sin (2 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

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

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

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

Consolidate the answers.

$x = \frac{π n}{2}$ , for any integer $n$