Graph sin(theta)<0 , cot(theta)>0

$\sin (θ) < 0$ , $\cot (θ) > 0$

Step 1

Solve for $θ$ .

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

$θ < \arcsin (0)$

Simplify the right side.

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

$θ < 0$

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.

$θ = π - 0$

Subtract $0$ from $π$ .

$θ = π$

Find the period of $\sin (θ)$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

Consolidate the answers.

$θ = π n$ , for any integer $n$

Use each root to create test intervals.

$0 < θ < π$

$π < θ < 2 π$

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

Test a value on the interval $0 < θ < π$ to see if it makes the inequality true.

Choose a value on the interval $0 < θ < π$ and see if this value makes the original inequality true.

$θ = 2$

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

$\sin (2) < 0$

The left side $0.90929742$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $π < θ < 2 π$ to see if it makes the inequality true.

Choose a value on the interval $π < θ < 2 π$ and see if this value makes the original inequality true.

$θ = 5$

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

$\sin (5) < 0$

The left side $- 0.95892427$ is less than the right side $0$ , which means that the given statement is always true.

True

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

$0 < θ < π$ False

$π < θ < 2 π$ True

$0 < θ < π$ False

$π < θ < 2 π$ True

The solution consists of all of the true intervals.

$π + 2 π n < θ < 2 π + 2 π n$ , for any integer $n$

Step 2

Solve for $θ$ .

Take the inverse cotangent of both sides of the equation to extract $θ$ from inside the cotangent.

$θ > arccot (0)$

Simplify the right side.

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

$θ > \frac{π}{2}$

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$θ = π + \frac{π}{2}$

Simplify $π + \frac{π}{2}$ .

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

$θ = π \cdot \frac{2}{2} + \frac{π}{2}$

Combine fractions.

Combine $π$ and $\frac{2}{2}$ .

$θ = \frac{π \cdot 2}{2} + \frac{π}{2}$

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 2 + π}{2}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$θ = \frac{2 \cdot π + π}{2}$

Add $2 π$ and $π$ .

$θ = \frac{3 π}{2}$

Find the period of $\cot (θ)$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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

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

Consolidate the answers.

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

Use each root to create test intervals.

$\frac{π}{2} < θ < \frac{3 π}{2}$

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

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

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

$θ = 3$

Replace $θ$ with $3$ in the original inequality.

$\cot (3) > 0$

The left side $- 7.01525255$ is not greater than the right side $0$ , which means that the given statement is false.

False

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

$\frac{π}{2} < θ < \frac{3 π}{2}$ False

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 3

Plot each graph on the same coordinate system.

$\sin (θ) < 0$

$\cot (θ) > 0$

Step 4

Graph sin(theta)<0 , cot(theta)>0

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