Trigonometry Examples

$5 \sin (x) = - 3$

Step 1

Divide each term in $5 \sin (x) = - 3$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \sin (x) = - 3$ by $5$ .

$\frac{5 \sin (x)}{5} = \frac{- 3}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \sin (x)}{5} = \frac{- 3}{5}$

Step 1.2.1.2

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

$\sin (x) = \frac{- 3}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{3}{5}$

Step 2

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

$x = \arcsin (- \frac{3}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (- \frac{3}{5})$ .

$x = - 0.6435011$

Step 4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 0.6435011 + 3.14159265$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $2 π$ from $2 (3.14159265) + 0.6435011 + 3.14159265$ .

$x = 2 (3.14159265) + 0.6435011 + 3.14159265 - 2 π$

Step 5.2

The resulting angle of $3.78509376$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.6435011 + 3.14159265$ .

$x = 3.78509376$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

Add $2 π$ to every negative angle to get positive angles.

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Step 7.1

Add $2 π$ to $- 0.6435011$ to find the positive angle.

$- 0.6435011 + 2 π$

Step 7.2

Subtract $0.6435011$ from $2 π$ .

$5.63968419$

Step 7.3

List the new angles.

$x = 5.63968419$

Step 8

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

$x = 3.78509376 + 2 π n, 5.63968419 + 2 π n$ , for any integer $n$