Find All Complex Solutions tan(theta)=- square root of 2sin(theta)

$\tan (θ) = - \sqrt{2} \sin (θ)$

Step 1

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

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

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

Simplify the left side.

Cancel the common factor of $\tan (θ)$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

Separate fractions.

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

Rewrite $\tan (θ)$ in terms of sines and cosines.

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

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (θ)}{\cos (θ)}$ .

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

Write $\sin (θ)$ as a fraction with denominator $1$ .

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

Cancel the common factor of $\sin (θ)$ .

Cancel the common factor.

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

Rewrite the expression.

$1 = \frac{- \sqrt{2}}{1} \cos (θ)$

Divide $- \sqrt{2}$ by $1$ .

$1 = - \sqrt{2} \cos (θ)$

Step 2

Rewrite the equation as $- \sqrt{2} \cos (θ) = 1$ .

$- \sqrt{2} \cos (θ) = 1$

Step 3

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

Cancel the common factor.

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

Divide $\cos (θ)$ by $1$ .

$\cos (θ) = \frac{1}{- \sqrt{2}}$

Simplify the right side.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{2}}$

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Step 4

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

$θ = \arccos (- \frac{\sqrt{2}}{2})$

Step 5

Simplify the right side.

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$θ = \frac{3 π}{4}$

Step 6

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

$θ = 2 π - \frac{3 π}{4}$

Step 7

Simplify $2 π - \frac{3 π}{4}$ .

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

$θ = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Combine fractions.

Combine $2 π$ and $\frac{4}{4}$ .

$θ = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 4 - 3 π}{4}$

Simplify the numerator.

Multiply $4$ by $2$ .

$θ = \frac{8 π - 3 π}{4}$

Subtract $3 π$ from $8 π$ .

$θ = \frac{5 π}{4}$

Step 8

Find the period of $\cos (θ)$ .

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 $1$ .

$2 π$

Step 9

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

$θ = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Find All Complex Solutions tan(theta)=- square root of 2sin(theta)

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