Find All Complex Solutions (cot(theta)- square root of 3)( square root of 2sin(theta)+1)=0

$(\cot (θ) - \sqrt{3}) (\sqrt{2} \sin (θ) + 1) = 0$

Step 1

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

$\cot (θ) - \sqrt{3} = 0$

$\sqrt{2} \sin (θ) + 1 = 0$

Step 2

Set $\cot (θ) - \sqrt{3}$ equal to $0$ and solve for $θ$ .

Set $\cot (θ) - \sqrt{3}$ equal to $0$ .

$\cot (θ) - \sqrt{3} = 0$

Solve $\cot (θ) - \sqrt{3} = 0$ for $θ$ .

Add $\sqrt{3}$ to both sides of the equation.

$\cot (θ) = \sqrt{3}$

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

$θ = arccot (\sqrt{3})$

Simplify the right side.

The exact value of $arccot (\sqrt{3})$ is $\frac{π}{6}$ .

$θ = \frac{π}{6}$

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.

$θ = π + \frac{π}{6}$

Simplify $π + \frac{π}{6}$ .

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$θ = π \cdot \frac{6}{6} + \frac{π}{6}$

Combine fractions.

Combine $π$ and $\frac{6}{6}$ .

$θ = \frac{π \cdot 6}{6} + \frac{π}{6}$

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 6 + π}{6}$

Simplify the numerator.

Move $6$ to the left of $π$ .

$θ = \frac{6 \cdot π + π}{6}$

Add $6 π$ and $π$ .

$θ = \frac{7 π}{6}$

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.

$θ = \frac{π}{6} + π n, \frac{7 π}{6} + π n$ , for any integer $n$

Step 3

Set $\sqrt{2} \sin (θ) + 1$ equal to $0$ and solve for $θ$ .

Set $\sqrt{2} \sin (θ) + 1$ equal to $0$ .

$\sqrt{2} \sin (θ) + 1 = 0$

Solve $\sqrt{2} \sin (θ) + 1 = 0$ for $θ$ .

Subtract $1$ from both sides of the equation.

$\sqrt{2} \sin (θ) = - 1$

Divide each term in $\sqrt{2} \sin (θ) = - 1$ by $\sqrt{2}$ and simplify.

Divide each term in $\sqrt{2} \sin (θ) = - 1$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \sin (θ)}{\sqrt{2}} = \frac{- 1}{\sqrt{2}}$

Simplify the left side.

Cancel the common factor of $\sqrt{2}$ .

Cancel the common factor.

$\frac{\sqrt{2} \sin (θ)}{\sqrt{2}} = \frac{- 1}{\sqrt{2}}$

Divide $\sin (θ)$ by $1$ .

$\sin (θ) = \frac{- 1}{\sqrt{2}}$

Simplify the right side.

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (θ) = - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (θ) = - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (θ) = - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (θ) = - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (θ) = - \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (θ) = - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (θ) = - \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sin (θ) = - \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\sin (θ) = - \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\sin (θ) = - \frac{\sqrt{2}}{2}$

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

$θ = \arcsin (- \frac{\sqrt{2}}{2})$

Simplify the right side.

The exact value of $\arcsin (- \frac{\sqrt{2}}{2})$ is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

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.

$θ = 2 π + \frac{π}{4} + π$

Simplify the expression to find the second solution.

Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$θ = 2 π + \frac{π}{4} + π - 2 π$

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$θ = \frac{5 π}{4}$

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 $1$ .

$2 π$

Add $2 π$ to every negative angle to get positive angles.

Add $2 π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + 2 π$

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 π \cdot \frac{4}{4} - \frac{π}{4}$

Combine fractions.

Combine $2 π$ and $\frac{4}{4}$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 4 - π}{4}$

Simplify the numerator.

Multiply $4$ by $2$ .

$\frac{8 π - π}{4}$

Subtract $π$ from $8 π$ .

$\frac{7 π}{4}$

List the new angles.

$θ = \frac{7 π}{4}$

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

$θ = \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 4

The final solution is all the values that make $(\cot (θ) - \sqrt{3}) (\sqrt{2} \sin (θ) + 1) = 0$ true.

$θ = \frac{π}{6} + π n, \frac{7 π}{6} + π n, \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 5

Consolidate $\frac{π}{6} + π n$ and $\frac{7 π}{6} + π n$ to $\frac{π}{6} + π n$ .

$θ = \frac{π}{6} + π n, \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Find All Complex Solutions (cot(theta)- square root of 3)( square root of 2sin(theta)+1)=0

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