Trigonometry Examples

$(\cot (x) + 1) \sin (x) = 0$

Step 1

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

$\cot (x) + 1 = 0$

$\sin (x) = 0$

Step 2

Set $\cot (x) + 1$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $\cot (x) + 1$ equal to $0$ .

$\cot (x) + 1 = 0$

Step 2.2

Solve $\cot (x) + 1 = 0$ for $x$ .

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$\cot (x) = - 1$

Step 2.2.2

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

$x = arccot (- 1)$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

The exact value of $arccot (- 1)$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 2.2.4

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

$x = \frac{3 π}{4} - π$

Step 2.2.5

Simplify the expression to find the second solution.

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Step 2.2.5.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$x = \frac{3 π}{4} - π + 2 π$

Step 2.2.5.2

The resulting angle of $\frac{7 π}{4}$ is positive and coterminal with $\frac{3 π}{4} - π$ .

$x = \frac{7 π}{4}$

Step 2.2.6

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

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Step 2.2.6.1

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

$\frac{π}{| b |}$

Step 2.2.6.2

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

$\frac{π}{| 1 |}$

Step 2.2.6.3

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

$\frac{π}{1}$

Step 2.2.6.4

Divide $π$ by $1$ .

$π$

Step 2.2.7

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

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 3

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

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Step 3.1

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

$\sin (x) = 0$

Step 3.2

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

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Step 3.2.1

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

$x = \arcsin (0)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

$x = 0$

Step 3.2.3

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 3.2.4

Subtract $0$ from $π$ .

$x = π$

Step 3.2.5

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

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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 $\sin (x)$ 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 4

The final solution is all the values that make $(\cot (x) + 1) \sin (x) = 0$ true.

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n, 2 π n, π + 2 π n$ , for any integer $n$

Step 5

Consolidate the answers.

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Step 5.1

Consolidate $\frac{3 π}{4} + π n$ and $\frac{7 π}{4} + π n$ to $\frac{3 π}{4} + π n$ .

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

Step 5.2

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

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

Step 6

Exclude the solutions that do not make $(\cot (x) + 1) \sin (x) = 0$ true.

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