Trigonometry Examples

$5 \sin (x) - 2 = 0$

Step 1

Add $2$ to both sides of the equation.

$5 \sin (x) = 2$

Step 2

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

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Step 2.1

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

$\frac{5 \sin (x)}{5} = \frac{2}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 \sin (x)}{5} = \frac{2}{5}$

Step 2.2.1.2

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

$\sin (x) = \frac{2}{5}$

Step 3

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

$x = \arcsin (\frac{2}{5})$

Step 4

Simplify the right side.

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Step 4.1

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

$x = 0.41151684$

Step 5

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.

$x = (3.14159265) - 0.41151684$

Step 6

Solve for $x$ .

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Step 6.1

Remove parentheses.

$x = 3.14159265 - 0.41151684$

Step 6.2

Remove parentheses.

$x = (3.14159265) - 0.41151684$

Step 6.3

Subtract $0.41151684$ from $3.14159265$ .

$x = 2.7300758$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

$x = 0.41151684 + 2 π n, 2.7300758 + 2 π n$ , for any integer $n$