Trigonometry Examples

$7 \sin (x) = \frac{7 π}{2}$

Step 1

Divide each term in $7 \sin (x) = \frac{7 π}{2}$ by $7$ and simplify.

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Step 1.1

Divide each term in $7 \sin (x) = \frac{7 π}{2}$ by $7$ .

$\frac{7 \sin (x)}{7} = \frac{\frac{7 π}{2}}{7}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $7$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{7 \sin (x)}{7} = \frac{\frac{7 π}{2}}{7}$

Step 1.2.1.2

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

$\sin (x) = \frac{\frac{7 π}{2}}{7}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (x) = \frac{7 π}{2} \cdot \frac{1}{7}$

Step 1.3.2

Cancel the common factor of $7$ .

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Step 1.3.2.1

Factor $7$ out of $7 π$ .

$\sin (x) = \frac{7 (π)}{2} \cdot \frac{1}{7}$

Step 1.3.2.2

Cancel the common factor.

$\sin (x) = \frac{7 π}{2} \cdot \frac{1}{7}$

Step 1.3.2.3

Rewrite the expression.

$\sin (x) = \frac{π}{2}$

Step 2

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

$x = \arcsin (\frac{π}{2})$

Step 3

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.

$x = π - \arcsin (\frac{π}{2})$

Step 4

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

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Step 4.1

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

$\frac{2 π}{| b |}$

Step 4.2

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

$\frac{2 π}{| 1 |}$

Step 4.3

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

$\frac{2 π}{1}$

Step 4.4

Divide $2 π$ by $1$ .

$2 π$

Step 5

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

$x = \arcsin (\frac{π}{2}) + 2 π n, π - \arcsin (\frac{π}{2}) + 2 π n$ , for any integer $n$