Solve for θ 2-sin(2theta)=4sin(2theta)

$2 - \sin (2 θ) = 4 \sin (2 θ)$

Step 1

Subtract $4 \sin (2 θ)$ from both sides of the equation.

$2 - \sin (2 θ) - 4 \sin (2 θ) = 0$

Step 2

Subtract $4 \sin (2 θ)$ from $- \sin (2 θ)$ .

$2 - 5 \sin (2 θ) = 0$

Step 3

Subtract $2$ from both sides of the equation.

$- 5 \sin (2 θ) = - 2$

Step 4

Divide each term in $- 5 \sin (2 θ) = - 2$ by $- 5$ and simplify.

Divide each term in $- 5 \sin (2 θ) = - 2$ by $- 5$ .

$\frac{- 5 \sin (2 θ)}{- 5} = \frac{- 2}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 \sin (2 θ)}{- 5} = \frac{- 2}{- 5}$

Divide $\sin (2 θ)$ by $1$ .

$\sin (2 θ) = \frac{- 2}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$\sin (2 θ) = \frac{2}{5}$

Step 5

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

$2 θ = \arcsin (\frac{2}{5})$

Step 6

Simplify the right side.

Evaluate $\arcsin (\frac{2}{5})$ .

$2 θ = 0.41151684$

Step 7

Divide each term in $2 θ = 0.41151684$ by $2$ and simplify.

Divide each term in $2 θ = 0.41151684$ by $2$ .

$\frac{2 θ}{2} = \frac{0.41151684}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{0.41151684}{2}$

Divide $θ$ by $1$ .

$θ = \frac{0.41151684}{2}$

Simplify the right side.

Divide $0.41151684$ by $2$ .

$θ = 0.20575842$

Step 8

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.

$2 θ = (3.14159265) - 0.41151684$

Step 9

Solve for $θ$ .

Subtract $0.41151684$ from $3.14159265$ .

$2 θ = 2.7300758$

Divide each term in $2 θ = 2.7300758$ by $2$ and simplify.

Divide each term in $2 θ = 2.7300758$ by $2$ .

$\frac{2 θ}{2} = \frac{2.7300758}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{2.7300758}{2}$

Divide $θ$ by $1$ .

$θ = \frac{2.7300758}{2}$

Simplify the right side.

Divide $2.7300758$ by $2$ .

$θ = 1.3650379$

Step 10

Find the period of $\sin (2 θ)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

Step 11

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

$θ = 0.20575842 + π n, 1.3650379 + π n$ , for any integer $n$

Solve for θ 2-sin(2theta)=4sin(2theta)

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