Trigonometry Examples

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

Step 1

Substitute $u$ for $\sin (2 x)$ .

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

Step 2

Add $1$ to both sides of the equation.

$u^{2} - 2 u + 1 = 0$

Step 3

Factor using the perfect square rule.

Tap for more steps...

Step 3.1

Rewrite $1$ as $1^{2}$ .

$u^{2} - 2 u + 1^{2} = 0$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 3.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 1 + 1^{2} = 0$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 1$ .

${(u - 1)}^{2} = 0$

Step 4

Set the $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 5

Add $1$ to both sides of the equation.

$u = 1$

Step 6

Substitute $\sin (2 x)$ for $u$ .

$\sin (2 x) = 1$

Step 7

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

$2 x = \arcsin (1)$

Step 8

Simplify the right side.

Tap for more steps...

Step 8.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$2 x = \frac{π}{2}$

Step 9

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

Tap for more steps...

Step 9.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 9.2

Simplify the left side.

Tap for more steps...

Step 9.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 9.3

Simplify the right side.

Tap for more steps...

Step 9.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 9.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 9.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 9.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 10

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.

$2 x = π - \frac{π}{2}$

Step 11

Solve for $x$ .

Tap for more steps...

Step 11.1

Simplify.

Tap for more steps...

Step 11.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 11.1.2

Combine $π$ and $\frac{2}{2}$ .

$2 x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 11.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 2 - π}{2}$

Step 11.1.4

Subtract $π$ from $π \cdot 2$ .

Tap for more steps...

Step 11.1.4.1

Reorder $π$ and $2$ .

$2 x = \frac{2 \cdot π - π}{2}$

Step 11.1.4.2

Subtract $π$ from $2 \cdot π$ .

$2 x = \frac{π}{2}$

Step 11.2

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

Tap for more steps...

Step 11.2.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 11.2.2

Simplify the left side.

Tap for more steps...

Step 11.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 11.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 11.2.3

Simplify the right side.

Tap for more steps...

Step 11.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 11.2.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 11.2.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 11.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 12

Find the period of $\sin (2 x)$ .

Tap for more steps...

Step 12.1

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

$\frac{2 π}{| b |}$

Step 12.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 12.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 12.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 12.4.2

Divide $π$ by $1$ .

$π$

Step 13

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

$x = \frac{π}{4} + π n$ , for any integer $n$