Trigonometry Examples

${(\frac{\sqrt{π}}{6})}^{2} + \sin (x) = 1$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{\sqrt{π}}{6}$ .

$\frac{{\sqrt{π}}^{2}}{6^{2}} + \sin (x) = 1$

Step 1.2

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{π}$ as $π^{\frac{1}{2}}$ .

$\frac{{(π^{\frac{1}{2}})}^{2}}{6^{2}} + \sin (x) = 1$

Step 1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{π^{\frac{1}{2} \cdot 2}}{6^{2}} + \sin (x) = 1$

Step 1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{π^{\frac{2}{2}}}{6^{2}} + \sin (x) = 1$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$\frac{π^{\frac{2}{2}}}{6^{2}} + \sin (x) = 1$

Step 1.2.4.2

Rewrite the expression.

$\frac{π^{1}}{6^{2}} + \sin (x) = 1$

Step 1.2.5

Simplify.

$\frac{π}{6^{2}} + \sin (x) = 1$

Step 1.3

Raise $6$ to the power of $2$ .

$\frac{π}{36} + \sin (x) = 1$

Step 2

Subtract $\frac{π}{36}$ from both sides of the equation.

$\sin (x) = 1 - \frac{π}{36}$

Step 3

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

$x = \arcsin (1 - \frac{π}{36})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (1 - \frac{π}{36})$ .

$x = 1.14992557$

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.

$x = (3.14159265) - 1.14992557$

Step 6

Solve for $x$ .

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Step 6.1

Remove parentheses.

$x = 3.14159265 - 1.14992557$

Step 6.2

Remove parentheses.

$x = (3.14159265) - 1.14992557$

Step 6.3

Subtract $1.14992557$ from $3.14159265$ .

$x = 1.99166707$

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.

$x = 1.14992557 + 2 π n, 1.99166707 + 2 π n$ , for any integer $n$