Trigonometry Examples

$49 \sin^{2} (x) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$49 \sin^{2} (x) = 1$

Step 2

Divide each term in $49 \sin^{2} (x) = 1$ by $49$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $49 \sin^{2} (x) = 1$ by $49$ .

$\frac{49 \sin^{2} (x)}{49} = \frac{1}{49}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $49$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{49 \sin^{2} (x)}{49} = \frac{1}{49}$

Step 2.2.1.2

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{1}{49}$

Step 3

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

$\sin (x) = \pm \sqrt{\frac{1}{49}}$

Step 4

Simplify $\pm \sqrt{\frac{1}{49}}$ .

Tap for more steps...

Step 4.1

Rewrite $\sqrt{\frac{1}{49}}$ as $\frac{\sqrt{1}}{\sqrt{49}}$ .

$\sin (x) = \pm \frac{\sqrt{1}}{\sqrt{49}}$

Step 4.2

Any root of $1$ is $1$ .

$\sin (x) = \pm \frac{1}{\sqrt{49}}$

Step 4.3

Simplify the denominator.

Tap for more steps...

Step 4.3.1

Rewrite $49$ as $7^{2}$ .

$\sin (x) = \pm \frac{1}{\sqrt{7^{2}}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\sin (x) = \pm \frac{1}{7}$

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

$\sin (x) = - \frac{1}{7}$

Step 5.3

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

$\sin (x) = \frac{1}{7}, - \frac{1}{7}$

Step 6

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

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

$\sin (x) = - \frac{1}{7}$

Step 7

Solve for $x$ in $\sin (x) = \frac{1}{7}$ .

Tap for more steps...

Step 7.1

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

$x = \arcsin (\frac{1}{7})$

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

Evaluate $\arcsin (\frac{1}{7})$ .

$x = 0.14334756$

Step 7.3

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.

$x = (3.14159265) - 0.14334756$

Step 7.4

Solve for $x$ .

Tap for more steps...

Step 7.4.1

Remove parentheses.

$x = 3.14159265 - 0.14334756$

Step 7.4.2

Remove parentheses.

$x = (3.14159265) - 0.14334756$

Step 7.4.3

Subtract $0.14334756$ from $3.14159265$ .

$x = 2.99824508$

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

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

$x = 0.14334756 + 2 π n, 2.99824508 + 2 π n$ , for any integer $n$

Step 8

Solve for $x$ in $\sin (x) = - \frac{1}{7}$ .

Tap for more steps...

Step 8.1

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

$x = \arcsin (- \frac{1}{7})$

Step 8.2

Simplify the right side.

Tap for more steps...

Step 8.2.1

Evaluate $\arcsin (- \frac{1}{7})$ .

$x = - 0.14334756$

Step 8.3

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.

$x = 2 (3.14159265) + 0.14334756 + 3.14159265$

Step 8.4

Simplify the expression to find the second solution.

Tap for more steps...

Step 8.4.1

Subtract $2 π$ from $2 (3.14159265) + 0.14334756 + 3.14159265$ .

$x = 2 (3.14159265) + 0.14334756 + 3.14159265 - 2 π$

Step 8.4.2

The resulting angle of $3.28494022$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.14334756 + 3.14159265$ .

$x = 3.28494022$

Step 8.5

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

Tap for more steps...

Step 8.5.1

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

$\frac{2 π}{| b |}$

Step 8.5.2

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

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

$\frac{2 π}{1}$

Step 8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.6

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

Tap for more steps...

Step 8.6.1

Add $2 π$ to $- 0.14334756$ to find the positive angle.

$- 0.14334756 + 2 π$

Step 8.6.2

Subtract $0.14334756$ from $2 π$ .

$6.13983773$

Step 8.6.3

List the new angles.

$x = 6.13983773$

Step 8.7

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

$x = 3.28494022 + 2 π n, 6.13983773 + 2 π n$ , for any integer $n$

Step 9

List all of the solutions.

$x = 0.14334756 + 2 π n, 2.99824508 + 2 π n, 3.28494022 + 2 π n, 6.13983773 + 2 π n$ , for any integer $n$

Step 10

Consolidate the solutions.

Tap for more steps...

Step 10.1

Consolidate $0.14334756 + 2 π n$ and $3.28494022 + 2 π n$ to $0.14334756 + π n$ .

$x = 0.14334756 + π n, 2.99824508 + 2 π n, 6.13983773 + 2 π n$ , for any integer $n$

Step 10.2

Consolidate $2.99824508 + 2 π n$ and $6.13983773 + 2 π n$ to $2.99824508 + π n$ .

$x = 0.14334756 + π n, 2.99824508 + π n$ , for any integer $n$