Trigonometry Examples

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

Step 1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (x) = 0$

$\cos (x) = 0$

Step 2

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 2.2

Solve $\sin (x) = 0$ for $x$ .

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Step 2.2.1

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

$x = \arcsin (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

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

$x = 0$

Step 2.2.3

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.

$x = π - 0$

Step 2.2.4

Subtract $0$ from $π$ .

$x = π$

Step 2.2.5

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

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Step 2.2.5.1

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

$\frac{2 π}{| b |}$

Step 2.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.2.5.3

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

$\frac{2 π}{1}$

Step 2.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.2.6

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

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 3

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 3.2

Solve $\cos (x) = 0$ for $x$ .

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Step 3.2.1

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

$x = \arccos (0)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 3.2.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 3.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 3.2.4.1

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

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

Step 3.2.4.2

Combine fractions.

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Step 3.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 3.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 3.2.4.3

Simplify the numerator.

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Step 3.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 3.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 3.2.5

Find the period of $\cos (x)$ .

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Step 3.2.5.1

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

$\frac{2 π}{| b |}$

Step 3.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 3.2.5.3

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

$\frac{2 π}{1}$

Step 3.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2.6

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

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

Step 4

The final solution is all the values that make $\sin (x) \cos (x) > 0$ true.

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

Step 5

Consolidate the answers.

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

Step 6

Use each root to create test intervals.

$0 < x < \frac{π}{2}$

$\frac{π}{2} < x < π$

$π < x < \frac{3 π}{2}$

$\frac{3 π}{2} < x < 2 π$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $0 < x < \frac{π}{2}$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $0 < x < \frac{π}{2}$ and see if this value makes the original inequality true.

$x = 1$

Step 7.1.2

Replace $x$ with $1$ in the original inequality.

$(\sin (1)) (\cos (1)) > 0$

Step 7.1.3

The left side $0.45464871$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $\frac{π}{2} < x < π$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $\frac{π}{2} < x < π$ and see if this value makes the original inequality true.

$x = 2$

Step 7.2.2

Replace $x$ with $2$ in the original inequality.

$(\sin (2)) (\cos (2)) > 0$

Step 7.2.3

The left side $- 0.37840124$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $π < x < \frac{3 π}{2}$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $π < x < \frac{3 π}{2}$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

Replace $x$ with $4$ in the original inequality.

$(\sin (4)) (\cos (4)) > 0$

Step 7.3.3

The left side $0.49467912$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

Test a value on the interval $\frac{3 π}{2} < x < 2 π$ to see if it makes the inequality true.

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Step 7.4.1

Choose a value on the interval $\frac{3 π}{2} < x < 2 π$ and see if this value makes the original inequality true.

$x = 5$

Step 7.4.2

Replace $x$ with $5$ in the original inequality.

$(\sin (5)) (\cos (5)) > 0$

Step 7.4.3

The left side $- 0.27201055$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.5

Compare the intervals to determine which ones satisfy the original inequality.

$0 < x < \frac{π}{2}$ True

$\frac{π}{2} < x < π$ False

$π < x < \frac{3 π}{2}$ True

$\frac{3 π}{2} < x < 2 π$ False

$0 < x < \frac{π}{2}$ True

$\frac{π}{2} < x < π$ False

$π < x < \frac{3 π}{2}$ True

$\frac{3 π}{2} < x < 2 π$ False

Step 8

The solution consists of all of the true intervals.

$0 + 2 π n < x < \frac{π}{2} + 2 π n$ or $π + 2 π n < x < \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 9

Combine the intervals.

$2 π n < x < \frac{π}{2} + 2 π n or π + 2 π n < x < \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 10