Trigonometry Examples

$3 - 5 \sin (x) = 4 \sin (x) + 1$

Step 1

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

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Step 1.1

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

$3 - 5 \sin (x) - 4 \sin (x) = 1$

Step 1.2

Subtract $4 \sin (x)$ from $- 5 \sin (x)$ .

$3 - 9 \sin (x) = 1$

Step 2

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

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Step 2.1

Subtract $3$ from both sides of the equation.

$- 9 \sin (x) = 1 - 3$

Step 2.2

Subtract $3$ from $1$ .

$- 9 \sin (x) = - 2$

Step 3

Divide each term in $- 9 \sin (x) = - 2$ by $- 9$ and simplify.

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Step 3.1

Divide each term in $- 9 \sin (x) = - 2$ by $- 9$ .

$\frac{- 9 \sin (x)}{- 9} = \frac{- 2}{- 9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 9 \sin (x)}{- 9} = \frac{- 2}{- 9}$

Step 3.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 2}{- 9}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{2}{9}$

Step 4

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

$x = \arcsin (\frac{2}{9})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (\frac{2}{9})$ .

$x = 0.22409309$

Step 6

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.

$x = (3.14159265) - 0.22409309$

Step 7

Solve for $x$ .

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Step 7.1

Remove parentheses.

$x = 3.14159265 - 0.22409309$

Step 7.2

Remove parentheses.

$x = (3.14159265) - 0.22409309$

Step 7.3

Subtract $0.22409309$ from $3.14159265$ .

$x = 2.91749956$

Step 8

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

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 1 |}$

Step 8.3

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

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

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

$x = 0.22409309 + 2 π n, 2.91749956 + 2 π n$ , for any integer $n$