Trigonometry Examples

$2 + 2 \sin (x) = 4 \cos^{2} (x)$

Step 1

Subtract $2$ from both sides of the equation.

$2 \sin (x) = 4 \cos^{2} (x) - 2$

Step 2

Square both sides of the equation.

${(2 \sin (x))}^{2} = {(4 \cos^{2} (x) - 2)}^{2}$

Step 3

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

Tap for more steps...

Step 3.1

Apply the product rule to $2 \sin (x)$ .

$2^{2} \sin^{2} (x) = {(4 \cos^{2} (x) - 2)}^{2}$

Step 3.2

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

$4 \sin^{2} (x) = {(4 \cos^{2} (x) - 2)}^{2}$

Step 4

Simplify ${(4 \cos^{2} (x) - 2)}^{2}$ .

Tap for more steps...

Step 4.1

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

$4 \sin^{2} (x) = (4 \cos^{2} (x) - 2) (4 \cos^{2} (x) - 2)$

Step 4.2

Expand $(4 \cos^{2} (x) - 2) (4 \cos^{2} (x) - 2)$ using the FOIL Method.

Tap for more steps...

Step 4.2.1

Apply the distributive property.

$4 \sin^{2} (x) = 4 \cos^{2} (x) (4 \cos^{2} (x) - 2) - 2 (4 \cos^{2} (x) - 2)$

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

Tap for more steps...

Step 4.3.1

Simplify each term.

Tap for more steps...

Step 4.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.2

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

Tap for more steps...

Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

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

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

Step 4.3.1.2.3

Add $2$ and $2$ .

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

Step 4.3.1.3

Multiply $4$ by $4$ .

$4 \sin^{2} (x) = 16 \cos^{4} (x) + 4 \cos^{2} (x) \cdot - 2 - 2 (4 \cos^{2} (x)) - 2 \cdot - 2$

Step 4.3.1.4

Multiply $- 2$ by $4$ .

$4 \sin^{2} (x) = 16 \cos^{4} (x) - 8 \cos^{2} (x) - 2 (4 \cos^{2} (x)) - 2 \cdot - 2$

Step 4.3.1.5

Multiply $4$ by $- 2$ .

$4 \sin^{2} (x) = 16 \cos^{4} (x) - 8 \cos^{2} (x) - 8 \cos^{2} (x) - 2 \cdot - 2$

Step 4.3.1.6

Multiply $- 2$ by $- 2$ .

$4 \sin^{2} (x) = 16 \cos^{4} (x) - 8 \cos^{2} (x) - 8 \cos^{2} (x) + 4$

Step 4.3.2

Subtract $8 \cos^{2} (x)$ from $- 8 \cos^{2} (x)$ .

$4 \sin^{2} (x) = 16 \cos^{4} (x) - 16 \cos^{2} (x) + 4$

Step 5

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 5.1

Subtract $16 \cos^{4} (x)$ from both sides of the equation.

$4 \sin^{2} (x) - 16 \cos^{4} (x) = - 16 \cos^{2} (x) + 4$

Step 5.2

Add $16 \cos^{2} (x)$ to both sides of the equation.

$4 \sin^{2} (x) - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 4$

Step 5.3

Subtract $4$ from both sides of the equation.

$4 \sin^{2} (x) - 16 \cos^{4} (x) + 16 \cos^{2} (x) - 4 = 0$

Step 6

Simplify $4 \sin^{2} (x) - 16 \cos^{4} (x) + 16 \cos^{2} (x) - 4$ .

Tap for more steps...

Step 6.1

Move $- 4$ .

$4 \sin^{2} (x) - 4 - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 0$

Step 6.2

Reorder $4 \sin^{2} (x)$ and $- 4$ .

$- 4 + 4 \sin^{2} (x) - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 0$

Step 6.3

Factor $- 4$ out of $- 4$ .

$- 4 (1) + 4 \sin^{2} (x) - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 0$

Step 6.4

Factor $- 4$ out of $4 \sin^{2} (x)$ .

$- 4 (1) - 4 (- \sin^{2} (x)) - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 0$

Step 6.5

Factor $- 4$ out of $- 4 (1) - 4 (- \sin^{2} (x))$ .

$- 4 (1 - \sin^{2} (x)) - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 0$

Step 6.6

Apply pythagorean identity.

$- 4 \cos^{2} (x) - 16 \cos^{4} (x) + 16 \cos^{2} (x) = 0$

Step 6.7

Add $- 4 \cos^{2} (x)$ and $16 \cos^{2} (x)$ .

$12 \cos^{2} (x) - 16 \cos^{4} (x) = 0$

Step 7

Solve for $x$ .

Tap for more steps...

Step 7.1

Factor the left side of the equation.

Tap for more steps...

Step 7.1.1

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

$12 \cos^{2} (x) - 16 {(\cos^{2} (x))}^{2} = 0$

Step 7.1.2

Let $u = \cos^{2} (x)$ . Substitute $u$ for all occurrences of $\cos^{2} (x)$ .

$12 u - 16 u^{2} = 0$

Step 7.1.3

Factor $4 u$ out of $12 u - 16 u^{2}$ .

Tap for more steps...

Step 7.1.3.1

Factor $4 u$ out of $12 u$ .

$4 u (3) - 16 u^{2} = 0$

Step 7.1.3.2

Factor $4 u$ out of $- 16 u^{2}$ .

$4 u (3) + 4 u (- 4 u) = 0$

Step 7.1.3.3

Factor $4 u$ out of $4 u (3) + 4 u (- 4 u)$ .

$4 u (3 - 4 u) = 0$

Step 7.1.4

Replace all occurrences of $u$ with $\cos^{2} (x)$ .

$4 \cos^{2} (x) (3 - 4 \cos^{2} (x)) = 0$

Step 7.2

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

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

$3 - 4 \cos^{2} (x) = 0$

Step 7.3

Set $\cos^{2} (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.3.1

Set $\cos^{2} (x)$ equal to $0$ .

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

Step 7.3.2

Solve $\cos^{2} (x) = 0$ for $x$ .

Tap for more steps...

Step 7.3.2.1

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

$\cos (x) = \pm \sqrt{0}$

Step 7.3.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 7.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$\cos (x) = \pm \sqrt{0^{2}}$

Step 7.3.2.2.2

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

$\cos (x) = \pm 0$

Step 7.3.2.2.3

Plus or minus $0$ is $0$ .

$\cos (x) = 0$

Step 7.3.2.3

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

$x = \arccos (0)$

Step 7.3.2.4

Simplify the right side.

Tap for more steps...

Step 7.3.2.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 7.3.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 7.3.2.6

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 7.3.2.6.1

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

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

Step 7.3.2.6.2

Combine fractions.

Tap for more steps...

Step 7.3.2.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 7.3.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 7.3.2.6.3

Simplify the numerator.

Tap for more steps...

Step 7.3.2.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 7.3.2.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 7.3.2.7

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

Tap for more steps...

Step 7.3.2.7.1

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

$\frac{2 π}{| b |}$

Step 7.3.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3.2.7.3

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

$\frac{2 π}{1}$

Step 7.3.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.3.2.8

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

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

Step 7.4

Set $3 - 4 \cos^{2} (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.4.1

Set $3 - 4 \cos^{2} (x)$ equal to $0$ .

$3 - 4 \cos^{2} (x) = 0$

Step 7.4.2

Solve $3 - 4 \cos^{2} (x) = 0$ for $x$ .

Tap for more steps...

Step 7.4.2.1

Subtract $3$ from both sides of the equation.

$- 4 \cos^{2} (x) = - 3$

Step 7.4.2.2

Divide each term in $- 4 \cos^{2} (x) = - 3$ by $- 4$ and simplify.

Tap for more steps...

Step 7.4.2.2.1

Divide each term in $- 4 \cos^{2} (x) = - 3$ by $- 4$ .

$\frac{- 4 \cos^{2} (x)}{- 4} = \frac{- 3}{- 4}$

Step 7.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 7.4.2.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 7.4.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 \cos^{2} (x)}{- 4} = \frac{- 3}{- 4}$

Step 7.4.2.2.2.1.2

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

$\cos^{2} (x) = \frac{- 3}{- 4}$

Step 7.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 7.4.2.2.3.1

Dividing two negative values results in a positive value.

$\cos^{2} (x) = \frac{3}{4}$

Step 7.4.2.3

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

$\cos (x) = \pm \sqrt{\frac{3}{4}}$

Step 7.4.2.4

Simplify $\pm \sqrt{\frac{3}{4}}$ .

Tap for more steps...

Step 7.4.2.4.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\cos (x) = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Step 7.4.2.4.2

Simplify the denominator.

Tap for more steps...

Step 7.4.2.4.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.4.2.4.2.2

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

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

Step 7.4.2.5

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

Tap for more steps...

Step 7.4.2.5.1

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

$\cos (x) = \frac{\sqrt{3}}{2}$

Step 7.4.2.5.2

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

$\cos (x) = - \frac{\sqrt{3}}{2}$

Step 7.4.2.5.3

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

$\cos (x) = \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}$

Step 7.4.2.6

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

$\cos (x) = \frac{\sqrt{3}}{2}$

$\cos (x) = - \frac{\sqrt{3}}{2}$

Step 7.4.2.7

Solve for $x$ in $\cos (x) = \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 7.4.2.7.1

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

$x = \arccos (\frac{\sqrt{3}}{2})$

Step 7.4.2.7.2

Simplify the right side.

Tap for more steps...

Step 7.4.2.7.2.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 7.4.2.7.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{6}$

Step 7.4.2.7.4

Simplify $2 π - \frac{π}{6}$ .

Tap for more steps...

Step 7.4.2.7.4.1

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

$x = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.4.2.7.4.2

Combine fractions.

Tap for more steps...

Step 7.4.2.7.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 7.4.2.7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 7.4.2.7.4.3

Simplify the numerator.

Tap for more steps...

Step 7.4.2.7.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 7.4.2.7.4.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 7.4.2.7.5

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

Tap for more steps...

Step 7.4.2.7.5.1

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

$\frac{2 π}{| b |}$

Step 7.4.2.7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.4.2.7.5.3

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

$\frac{2 π}{1}$

Step 7.4.2.7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.4.2.7.6

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

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

Step 7.4.2.8

Solve for $x$ in $\cos (x) = - \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 7.4.2.8.1

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

$x = \arccos (- \frac{\sqrt{3}}{2})$

Step 7.4.2.8.2

Simplify the right side.

Tap for more steps...

Step 7.4.2.8.2.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

Step 7.4.2.8.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{5 π}{6}$

Step 7.4.2.8.4

Simplify $2 π - \frac{5 π}{6}$ .

Tap for more steps...

Step 7.4.2.8.4.1

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

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 7.4.2.8.4.2

Combine fractions.

Tap for more steps...

Step 7.4.2.8.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 7.4.2.8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 7.4.2.8.4.3

Simplify the numerator.

Tap for more steps...

Step 7.4.2.8.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 7.4.2.8.4.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 7.4.2.8.5

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

Tap for more steps...

Step 7.4.2.8.5.1

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

$\frac{2 π}{| b |}$

Step 7.4.2.8.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.4.2.8.5.3

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

$\frac{2 π}{1}$

Step 7.4.2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.4.2.8.6

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

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

Step 7.4.2.9

List all of the solutions.

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

Step 7.4.2.10

Consolidate the solutions.

Tap for more steps...

Step 7.4.2.10.1

Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

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

Step 7.4.2.10.2

Consolidate $\frac{11 π}{6} + 2 π n$ and $\frac{5 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

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

Step 7.5

The final solution is all the values that make $4 \cos^{2} (x) (3 - 4 \cos^{2} (x)) = 0$ true.

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

Step 8

Consolidate the answers.

Tap for more steps...

Step 8.1

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

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

Step 8.2

Consolidate the answers.

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

Step 9

Verify each of the solutions by substituting them into $2 + 2 \sin (x) = 4 \cos^{2} (x)$ and solving.

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