Trigonometry Examples

$11 \sin (θ) + 2 = 4 \sin (θ) + 2$

Step 1

Move all terms containing $\sin (θ)$ to the left side of the equation.

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Step 1.1

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

$11 \sin (θ) + 2 - 4 \sin (θ) = 2$

Step 1.2

Subtract $4 \sin (θ)$ from $11 \sin (θ)$ .

$7 \sin (θ) + 2 = 2$

Step 2

Move all terms not containing $\sin (θ)$ to the right side of the equation.

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Step 2.1

Subtract $2$ from both sides of the equation.

$7 \sin (θ) = 2 - 2$

Step 2.2

Subtract $2$ from $2$ .

$7 \sin (θ) = 0$

Step 3

Divide each term in $7 \sin (θ) = 0$ by $7$ and simplify.

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Step 3.1

Divide each term in $7 \sin (θ) = 0$ by $7$ .

$\frac{7 \sin (θ)}{7} = \frac{0}{7}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{7 \sin (θ)}{7} = \frac{0}{7}$

Step 3.2.1.2

Divide $\sin (θ)$ by $1$ .

$\sin (θ) = \frac{0}{7}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $7$ .

$\sin (θ) = 0$

Step 4

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

$θ = \arcsin (0)$

Step 5

Simplify the right side.

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Step 5.1

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

$θ = 0$

Step 6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$θ = 180 - 0$

Step 7

Subtract $0$ from $180$ .

$θ = 180$

Step 8

Find the period of $\sin (θ)$ .

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Step 8.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 8.2

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

$\frac{360}{| 1 |}$

Step 8.3

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

$\frac{360}{1}$

Step 8.4

Divide $360$ by $1$ .

$360$

Step 9

The period of the $\sin (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 360 n, 180 + 360 n$ , for any integer $n$

Step 10

Consolidate the answers.

$θ = 180 n$ , for any integer $n$