Trigonometry Examples

$\sin^{2} (x) - \frac{1}{3} = 0$

Step 1

Add $\frac{1}{3}$ to both sides of the equation.

$\sin^{2} (x) = \frac{1}{3}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{\frac{1}{3}}$

Step 3

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Tap for more steps...

Step 3.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\sin (x) = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 3.2

Any root of $1$ is $1$ .

$\sin (x) = \pm \frac{1}{\sqrt{3}}$

Step 3.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (x) = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (x) = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.5

Add $1$ and $1$ .

$\sin (x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sin (x) = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.6.2

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

$\sin (x) = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

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

$\sin (x) = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.6.4.1

Cancel the common factor.

$\sin (x) = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.6.4.2

Rewrite the expression.

$\sin (x) = \pm \frac{\sqrt{3}}{3^{1}}$

Step 3.4.6.5

Evaluate the exponent.

$\sin (x) = \pm \frac{\sqrt{3}}{3}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$\sin (x) = \frac{\sqrt{3}}{3}$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 5

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{\sqrt{3}}{3}$

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

Step 6

Solve for $x$ in $\sin (x) = \frac{\sqrt{3}}{3}$ .

Tap for more steps...

Step 6.1

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

$x = \arcsin (\frac{\sqrt{3}}{3})$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Evaluate $\arcsin (\frac{\sqrt{3}}{3})$ .

$x = 0.6154797$

Step 6.3

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.

$x = (3.14159265) - 0.6154797$

Step 6.4

Solve for $x$ .

Tap for more steps...

Step 6.4.1

Remove parentheses.

$x = 3.14159265 - 0.6154797$

Step 6.4.2

Remove parentheses.

$x = (3.14159265) - 0.6154797$

Step 6.4.3

Subtract $0.6154797$ from $3.14159265$ .

$x = 2.52611294$

Step 6.5

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

Tap for more steps...

Step 6.5.1

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

$\frac{2 π}{| b |}$

Step 6.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.5.3

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

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

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

$x = 0.6154797 + 2 π n, 2.52611294 + 2 π n$ , for any integer $n$

Step 7

Solve for $x$ in $\sin (x) = - \frac{\sqrt{3}}{3}$ .

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

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

$x = - 0.6154797$

Step 7.3

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.

$x = 2 (3.14159265) + 0.6154797 + 3.14159265$

Step 7.4

Simplify the expression to find the second solution.

Tap for more steps...

Step 7.4.1

Subtract $2 π$ from $2 (3.14159265) + 0.6154797 + 3.14159265$ .

$x = 2 (3.14159265) + 0.6154797 + 3.14159265 - 2 π$

Step 7.4.2

The resulting angle of $3.75707236$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.6154797 + 3.14159265$ .

$x = 3.75707236$

Step 7.5

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

Tap for more steps...

Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

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

Tap for more steps...

Step 7.6.1

Add $2 π$ to $- 0.6154797$ to find the positive angle.

$- 0.6154797 + 2 π$

Step 7.6.2

Subtract $0.6154797$ from $2 π$ .

$5.66770559$

Step 7.6.3

List the new angles.

$x = 5.66770559$

Step 7.7

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

$x = 3.75707236 + 2 π n, 5.66770559 + 2 π n$ , for any integer $n$

Step 8

List all of the solutions.

$x = 0.6154797 + 2 π n, 2.52611294 + 2 π n, 3.75707236 + 2 π n, 5.66770559 + 2 π n$ , for any integer $n$

Step 9

Consolidate the solutions.

Tap for more steps...

Step 9.1

Consolidate $0.6154797 + 2 π n$ and $3.75707236 + 2 π n$ to $0.6154797 + π n$ .

$x = 0.6154797 + π n, 2.52611294 + 2 π n, 5.66770559 + 2 π n$ , for any integer $n$

Step 9.2

Consolidate $2.52611294 + 2 π n$ and $5.66770559 + 2 π n$ to $2.52611294 + π n$ .

$x = 0.6154797 + π n, 2.52611294 + π n$ , for any integer $n$