Trigonometry Examples

$9 \sin^{2} (x) - 5 \sin (x) = 1$

Step 1

Substitute $u$ for $\sin (x)$ .

$9 {(u)}^{2} - 5 u = 1$

Step 2

Subtract $1$ from both sides of the equation.

$9 u^{2} - 5 u - 1 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 9$ , $b = - 5$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (9 \cdot - 1)}}{2 \cdot 9}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise $- 5$ to the power of $2$ .

$u = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot - 1}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot - 1$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $9$ .

$u = \frac{5 \pm \sqrt{25 - 36 \cdot - 1}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $- 1$ .

$u = \frac{5 \pm \sqrt{25 + 36}}{2 \cdot 9}$

Step 5.1.3

Add $25$ and $36$ .

$u = \frac{5 \pm \sqrt{61}}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$u = \frac{5 \pm \sqrt{61}}{18}$

Step 6

The final answer is the combination of both solutions.

$u = \frac{5 + \sqrt{61}}{18}, \frac{5 - \sqrt{61}}{18}$

Step 7

Substitute $\sin (x)$ for $u$ .

$\sin (x) = \frac{5 + \sqrt{61}}{18}, \frac{5 - \sqrt{61}}{18}$

Step 8

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

$\sin (x) = \frac{5 + \sqrt{61}}{18}$

$\sin (x) = \frac{5 - \sqrt{61}}{18}$

Step 9

Solve for $x$ in $\sin (x) = \frac{5 + \sqrt{61}}{18}$ .

Tap for more steps...

Step 9.1

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

$x = \arcsin (\frac{5 + \sqrt{61}}{18})$

Step 9.2

Simplify the right side.

Tap for more steps...

Step 9.2.1

Evaluate $\arcsin (\frac{5 + \sqrt{61}}{18})$ .

$x = 0.79188753$

Step 9.3

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.

$x = (3.14159265) - 0.79188753$

Step 9.4

Solve for $x$ .

Tap for more steps...

Step 9.4.1

Remove parentheses.

$x = 3.14159265 - 0.79188753$

Step 9.4.2

Remove parentheses.

$x = (3.14159265) - 0.79188753$

Step 9.4.3

Subtract $0.79188753$ from $3.14159265$ .

$x = 2.34970512$

Step 9.5

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

Tap for more steps...

Step 9.5.1

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

$\frac{2 π}{| b |}$

Step 9.5.2

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

$\frac{2 π}{| 1 |}$

Step 9.5.3

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

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

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

$x = 0.79188753 + 2 π n, 2.34970512 + 2 π n$ , for any integer $n$

Step 10

Solve for $x$ in $\sin (x) = \frac{5 - \sqrt{61}}{18}$ .

Tap for more steps...

Step 10.1

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

$x = \arcsin (\frac{5 - \sqrt{61}}{18})$

Step 10.2

Simplify the right side.

Tap for more steps...

Step 10.2.1

Evaluate $\arcsin (\frac{5 - \sqrt{61}}{18})$ .

$x = - 0.15676629$

Step 10.3

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.

$x = (3.14159265) + 0.15676629$

Step 10.4

Solve for $x$ .

Tap for more steps...

Step 10.4.1

Remove parentheses.

$x = 3.14159265 + 0.15676629$

Step 10.4.2

Remove parentheses.

$x = (3.14159265) + 0.15676629$

Step 10.4.3

Add $3.14159265$ and $0.15676629$ .

$x = 3.29835895$

Step 10.5

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

Tap for more steps...

Step 10.5.1

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

$\frac{2 π}{| b |}$

Step 10.5.2

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

$\frac{2 π}{| 1 |}$

Step 10.5.3

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

$\frac{2 π}{1}$

Step 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

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

Tap for more steps...

Step 10.6.1

Add $2 π$ to $- 0.15676629$ to find the positive angle.

$- 0.15676629 + 2 π$

Step 10.6.2

Subtract $0.15676629$ from $2 π$ .

$6.126419$

Step 10.6.3

List the new angles.

$x = 6.126419$

Step 10.7

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

$x = 3.29835895 + 2 π n, 6.126419 + 2 π n$ , for any integer $n$

Step 11

List all of the solutions.

$x = 0.79188753 + 2 π n, 2.34970512 + 2 π n, 3.29835895 + 2 π n, 6.126419 + 2 π n$ , for any integer $n$