Trigonometry Examples

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Square both sides of the equation.

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

Step 3

Apply the cosine double-angle identity.

$\sin^{2} (x) = \cos^{2} (2 x)$

Step 4

Subtract $\cos^{2} (2 x)$ from both sides of the equation.

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

Step 5

Simplify the left side of the equation.

Tap for more steps...

Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \cos (2 x)$ .

$(\sin (x) + \cos (2 x)) (\sin (x) - \cos (2 x)) = 0$

Step 5.2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 5.3

Simplify each term.

Tap for more steps...

Step 5.3.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Multiply $- 1$ by $1$ .

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

Step 5.3.4

Multiply $- 2$ by $- 1$ .

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

Step 5.4

Expand $(\sin (x) + 1 - 2 \sin^{2} (x)) (\sin (x) - 1 + 2 \sin^{2} (x))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.5

Simplify terms.

Tap for more steps...

Step 5.5.1

Combine the opposite terms in $\sin (x) \sin (x) + \sin (x) \cdot - 1 + \sin (x) (2 \sin^{2} (x)) + 1 \sin (x) + 1 \cdot - 1 + 1 (2 \sin^{2} (x)) - 2 \sin^{2} (x) \sin (x) - 2 \sin^{2} (x) \cdot - 1 - 2 \sin^{2} (x) (2 \sin^{2} (x))$ .

Tap for more steps...

Step 5.5.1.1

Reorder the factors in the terms $\sin (x) (2 \sin^{2} (x))$ and $- 2 \sin^{2} (x) \sin (x)$ .

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

Step 5.5.1.2

Subtract $2 \sin^{2} (x) \sin (x)$ from $2 \sin^{2} (x) \sin (x)$ .

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

Step 5.5.1.3

Add $\sin (x) \sin (x) + \sin (x) \cdot - 1 + 1 \sin (x) + 1 \cdot - 1 + 1 (2 \sin^{2} (x))$ and $0$ .

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

Step 5.5.2

Simplify each term.

Tap for more steps...

Step 5.5.2.1

Multiply $\sin (x) \sin (x)$ .

Tap for more steps...

Step 5.5.2.1.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 5.5.2.1.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 5.5.2.1.3

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

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

Step 5.5.2.1.4

Add $1$ and $1$ .

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

Step 5.5.2.2

Move $- 1$ to the left of $\sin (x)$ .

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

Step 5.5.2.3

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

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

Step 5.5.2.4

Multiply $\sin (x)$ by $1$ .

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

Step 5.5.2.5

Multiply $- 1$ by $1$ .

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

Step 5.5.2.6

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

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

Step 5.5.2.7

Multiply $- 1$ by $- 2$ .

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

Step 5.5.2.8

Rewrite using the commutative property of multiplication.

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

Step 5.5.2.9

Multiply $\sin^{2} (x)$ by $\sin^{2} (x)$ by adding the exponents.

Tap for more steps...

Step 5.5.2.9.1

Move $\sin^{2} (x)$ .

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

Step 5.5.2.9.2

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

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

Step 5.5.2.9.3

Add $2$ and $2$ .

$\sin^{2} (x) - \sin (x) + \sin (x) - 1 + 2 \sin^{2} (x) + 2 \sin^{2} (x) - 2 \cdot (2 \sin^{4} (x)) = 0$

Step 5.5.2.10

Multiply $- 2$ by $2$ .

$\sin^{2} (x) - \sin (x) + \sin (x) - 1 + 2 \sin^{2} (x) + 2 \sin^{2} (x) - 4 \sin^{4} (x) = 0$

Step 5.5.3

Simplify by adding terms.

Tap for more steps...

Step 5.5.3.1

Combine the opposite terms in $\sin^{2} (x) - \sin (x) + \sin (x) - 1 + 2 \sin^{2} (x) + 2 \sin^{2} (x) - 4 \sin^{4} (x)$ .

Tap for more steps...

Step 5.5.3.1.1

Add $- \sin (x)$ and $\sin (x)$ .

$\sin^{2} (x) + 0 - 1 + 2 \sin^{2} (x) + 2 \sin^{2} (x) - 4 \sin^{4} (x) = 0$

Step 5.5.3.1.2

Add $\sin^{2} (x)$ and $0$ .

$\sin^{2} (x) - 1 + 2 \sin^{2} (x) + 2 \sin^{2} (x) - 4 \sin^{4} (x) = 0$

Step 5.5.3.2

Add $\sin^{2} (x)$ and $2 \sin^{2} (x)$ .

$- 1 + 3 \sin^{2} (x) + 2 \sin^{2} (x) - 4 \sin^{4} (x) = 0$

Step 5.5.3.3

Add $3 \sin^{2} (x)$ and $2 \sin^{2} (x)$ .

$- 1 + 5 \sin^{2} (x) - 4 \sin^{4} (x) = 0$

Step 6

Factor $- 1 + 5 \sin^{2} (x) - 4 \sin^{4} (x)$ .

Tap for more steps...

Step 6.1

Factor by grouping.

Tap for more steps...

Step 6.1.1

Reorder terms.

$- 4 \sin^{4} (x) + 5 \sin^{2} (x) - 1 = 0$

Step 6.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 4 \cdot - 1 = 4$ and whose sum is $b = 5$ .

Tap for more steps...

Step 6.1.2.1

Factor $5$ out of $5 \sin^{2} (x)$ .

$- 4 \sin^{4} (x) + 5 \sin^{2} (x) - 1 = 0$

Step 6.1.2.2

Rewrite $5$ as $1$ plus $4$

$- 4 \sin^{4} (x) + (1 + 4) \sin^{2} (x) - 1 = 0$

Step 6.1.2.3

Apply the distributive property.

$- 4 \sin^{4} (x) + 1 \sin^{2} (x) + 4 \sin^{2} (x) - 1 = 0$

Step 6.1.2.4

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

$- 4 \sin^{4} (x) + \sin^{2} (x) + 4 \sin^{2} (x) - 1 = 0$

Step 6.1.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 6.1.3.1

Group the first two terms and the last two terms.

$- 4 \sin^{4} (x) + \sin^{2} (x) + 4 \sin^{2} (x) - 1 = 0$

Step 6.1.3.2

Factor out the greatest common factor (GCF) from each group.

$\sin^{2} (x) (- 4 \sin^{2} (x) + 1) - (- 4 \sin^{2} (x) + 1) = 0$

Step 6.1.4

Factor the polynomial by factoring out the greatest common factor, $- 4 \sin^{2} (x) + 1$ .

$(- 4 \sin^{2} (x) + 1) (\sin^{2} (x) - 1) = 0$

Step 6.2

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

$(- 4 \sin^{2} (x) + 1^{2}) (\sin^{2} (x) - 1) = 0$

Step 6.3

Rewrite $4 \sin^{2} (x)$ as ${(2 \sin (x))}^{2}$ .

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

Step 6.4

Reorder $- {(2 \sin (x))}^{2}$ and $1^{2}$ .

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

Step 6.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 2 \sin (x)$ .

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

Step 6.6

Multiply $2$ by $- 1$ .

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

Step 6.7

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

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

Step 6.8

Factor.

Tap for more steps...

Step 6.8.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = 1$ .

$(1 + 2 \sin (x)) (1 - 2 \sin (x)) ((\sin (x) + 1) (\sin (x) - 1)) = 0$

Step 6.8.2

Remove unnecessary parentheses.

$(1 + 2 \sin (x)) (1 - 2 \sin (x)) (\sin (x) + 1) (\sin (x) - 1) = 0$

Step 7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$1 + 2 \sin (x) = 0$

$1 - 2 \sin (x) = 0$

$\sin (x) + 1 = 0$

$\sin (x) - 1 = 0$

Step 8

Set $1 + 2 \sin (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 8.1

Set $1 + 2 \sin (x)$ equal to $0$ .

$1 + 2 \sin (x) = 0$

Step 8.2

Solve $1 + 2 \sin (x) = 0$ for $x$ .

Tap for more steps...

Step 8.2.1

Subtract $1$ from both sides of the equation.

$2 \sin (x) = - 1$

Step 8.2.2

Divide each term in $2 \sin (x) = - 1$ by $2$ and simplify.

Tap for more steps...

Step 8.2.2.1

Divide each term in $2 \sin (x) = - 1$ by $2$ .

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

Step 8.2.2.2

Simplify the left side.

Tap for more steps...

Step 8.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 8.2.2.3

Simplify the right side.

Tap for more steps...

Step 8.2.2.3.1

Move the negative in front of the fraction.

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

Step 8.2.3

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

$x = \arcsin (- \frac{1}{2})$

Step 8.2.4

Simplify the right side.

Tap for more steps...

Step 8.2.4.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 8.2.5

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.

$x = 2 π + \frac{π}{6} + π$

Step 8.2.6

Simplify the expression to find the second solution.

Tap for more steps...

Step 8.2.6.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 8.2.6.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 8.2.7

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

Tap for more steps...

Step 8.2.7.1

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

$\frac{2 π}{| b |}$

Step 8.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 8.2.7.3

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

$\frac{2 π}{1}$

Step 8.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.2.8

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

Tap for more steps...

Step 8.2.8.1

Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 8.2.8.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 8.2.8.3

Combine fractions.

Tap for more steps...

Step 8.2.8.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 8.2.8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 8.2.8.4

Simplify the numerator.

Tap for more steps...

Step 8.2.8.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 8.2.8.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 8.2.8.5

List the new angles.

$x = \frac{11 π}{6}$

Step 8.2.9

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 9

Set $1 - 2 \sin (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 9.1

Set $1 - 2 \sin (x)$ equal to $0$ .

$1 - 2 \sin (x) = 0$

Step 9.2

Solve $1 - 2 \sin (x) = 0$ for $x$ .

Tap for more steps...

Step 9.2.1

Subtract $1$ from both sides of the equation.

$- 2 \sin (x) = - 1$

Step 9.2.2

Divide each term in $- 2 \sin (x) = - 1$ by $- 2$ and simplify.

Tap for more steps...

Step 9.2.2.1

Divide each term in $- 2 \sin (x) = - 1$ by $- 2$ .

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

Step 9.2.2.2

Simplify the left side.

Tap for more steps...

Step 9.2.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 9.2.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 9.2.2.3

Simplify the right side.

Tap for more steps...

Step 9.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 9.2.3

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

$x = \arcsin (\frac{1}{2})$

Step 9.2.4

Simplify the right side.

Tap for more steps...

Step 9.2.4.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 9.2.5

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.

$x = π - \frac{π}{6}$

Step 9.2.6

Simplify $π - \frac{π}{6}$ .

Tap for more steps...

Step 9.2.6.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 9.2.6.2

Combine fractions.

Tap for more steps...

Step 9.2.6.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 9.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 9.2.6.3

Simplify the numerator.

Tap for more steps...

Step 9.2.6.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 9.2.6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 9.2.7

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

Tap for more steps...

Step 9.2.7.1

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

$\frac{2 π}{| b |}$

Step 9.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 9.2.7.3

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

$\frac{2 π}{1}$

Step 9.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.2.8

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 10

Set $\sin (x) + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.1

Set $\sin (x) + 1$ equal to $0$ .

$\sin (x) + 1 = 0$

Step 10.2

Solve $\sin (x) + 1 = 0$ for $x$ .

Tap for more steps...

Step 10.2.1

Subtract $1$ from both sides of the equation.

$\sin (x) = - 1$

Step 10.2.2

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

$x = \arcsin (- 1)$

Step 10.2.3

Simplify the right side.

Tap for more steps...

Step 10.2.3.1

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

$x = - \frac{π}{2}$

Step 10.2.4

$x = 2 π + \frac{π}{2} + π$

Step 10.2.5

Simplify the expression to find the second solution.

Tap for more steps...

Step 10.2.5.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 10.2.5.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 10.2.6

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

Tap for more steps...

Step 10.2.6.1

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

$\frac{2 π}{| b |}$

Step 10.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 10.2.6.3

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

$\frac{2 π}{1}$

Step 10.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.2.7

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

Tap for more steps...

Step 10.2.7.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 10.2.7.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 10.2.7.3

Combine fractions.

Tap for more steps...

Step 10.2.7.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 10.2.7.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 10.2.7.4

Simplify the numerator.

Tap for more steps...

Step 10.2.7.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 10.2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 10.2.7.5

List the new angles.

$x = \frac{3 π}{2}$

Step 10.2.8

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

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 11

Set $\sin (x) - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 11.1

Set $\sin (x) - 1$ equal to $0$ .

$\sin (x) - 1 = 0$

Step 11.2

Solve $\sin (x) - 1 = 0$ for $x$ .

Tap for more steps...

Step 11.2.1

Add $1$ to both sides of the equation.

$\sin (x) = 1$

Step 11.2.2

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

$x = \arcsin (1)$

Step 11.2.3

Simplify the right side.

Tap for more steps...

Step 11.2.3.1

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

$x = \frac{π}{2}$

Step 11.2.4

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.

$x = π - \frac{π}{2}$

Step 11.2.5

Simplify $π - \frac{π}{2}$ .

Tap for more steps...

Step 11.2.5.1

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

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

Step 11.2.5.2

Combine fractions.

Tap for more steps...

Step 11.2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 11.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 11.2.5.3

Simplify the numerator.

Tap for more steps...

Step 11.2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 11.2.5.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 11.2.6

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

Tap for more steps...

Step 11.2.6.1

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

$\frac{2 π}{| b |}$

Step 11.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 11.2.6.3

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

$\frac{2 π}{1}$

Step 11.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.2.7

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

$x = \frac{π}{2} + 2 π n$ , for any integer $n$

Step 12

The final solution is all the values that make $(1 + 2 \sin (x)) (1 - 2 \sin (x)) (\sin (x) + 1) (\sin (x) - 1) = 0$ true.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{π}{2} + 2 π n$ , for any integer $n$

Step 13

Consolidate the answers.

$x = \frac{π}{6} + \frac{π n}{3}$ , for any integer $n$

Step 14

Verify each of the solutions by substituting them into $1 + \sin (x) = 2 \cos^{2} (x)$ and solving.

$x = \frac{π}{6} + \frac{2 π n}{3}$ , for any integer $n$