Trigonometry Examples

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

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\cos (x)}$ as ${\cos (x)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({\cos (x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({\cos (x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 \cos (x) - 1)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(2 \cos (x) - 1)}^{2}$ as $(2 \cos (x) - 1) (2 \cos (x) - 1)$ .

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

Step 2.3.1.2

Expand $(2 \cos (x) - 1) (2 \cos (x) - 1)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

$\cos (x) = 2 \cos (x) (2 \cos (x)) + 2 \cos (x) \cdot - 1 - 1 (2 \cos (x) - 1)$

Step 2.3.1.2.3

Apply the distributive property.

$\cos (x) = 2 \cos (x) (2 \cos (x)) + 2 \cos (x) \cdot - 1 - 1 (2 \cos (x)) - 1 \cdot - 1$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $2 \cos (x) (2 \cos (x))$ .

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Step 2.3.1.3.1.1.1

Multiply $2$ by $2$ .

$\cos (x) = 4 \cos (x) \cos (x) + 2 \cos (x) \cdot - 1 - 1 (2 \cos (x)) - 1 \cdot - 1$

Step 2.3.1.3.1.1.2

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

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

Step 2.3.1.3.1.1.3

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

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

Step 2.3.1.3.1.1.4

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

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

Step 2.3.1.3.1.1.5

Add $1$ and $1$ .

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

Step 2.3.1.3.1.2

Multiply $- 1$ by $2$ .

$\cos (x) = 4 \cos^{2} (x) - 2 \cos (x) - 1 (2 \cos (x)) - 1 \cdot - 1$

Step 2.3.1.3.1.3

Multiply $2$ by $- 1$ .

$\cos (x) = 4 \cos^{2} (x) - 2 \cos (x) - 2 \cos (x) - 1 \cdot - 1$

Step 2.3.1.3.1.4

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.3.2

Subtract $2 \cos (x)$ from $- 2 \cos (x)$ .

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

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4 \cos^{2} (x)$ from both sides of the equation.

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

Step 3.1.2

Add $4 \cos (x)$ to both sides of the equation.

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

Step 3.1.3

Subtract $1$ from both sides of the equation.

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

Step 3.2

Add $\cos (x)$ and $4 \cos (x)$ .

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

Step 3.3

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

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Step 3.3.1

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$5 u - 4 u^{2} - 1 = 0$

Step 3.3.2

Factor by grouping.

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Step 3.3.2.1

Reorder terms.

$- 4 u^{2} + 5 u - 1 = 0$

Step 3.3.2.2

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 $b = 5$ .

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Step 3.3.2.2.1

Factor $5$ out of $5 u$ .

$- 4 u^{2} + 5 (u) - 1 = 0$

Step 3.3.2.2.2

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

$- 4 u^{2} + (1 + 4) u - 1 = 0$

Step 3.3.2.2.3

Apply the distributive property.

$- 4 u^{2} + 1 u + 4 u - 1 = 0$

Step 3.3.2.2.4

Multiply $u$ by $1$ .

$- 4 u^{2} + u + 4 u - 1 = 0$

Step 3.3.2.3

Factor out the greatest common factor from each group.

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Step 3.3.2.3.1

Group the first two terms and the last two terms.

$(- 4 u^{2} + u) + 4 u - 1 = 0$

Step 3.3.2.3.2

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

$u (- 4 u + 1) - (- 4 u + 1) = 0$

Step 3.3.2.4

Factor the polynomial by factoring out the greatest common factor, $- 4 u + 1$ .

$(- 4 u + 1) (u - 1) = 0$

Step 3.3.3

Replace all occurrences of $u$ with $\cos (x)$ .

$(- 4 \cos (x) + 1) (\cos (x) - 1) = 0$

Step 3.4

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

$- 4 \cos (x) + 1 = 0$

$\cos (x) - 1 = 0$

Step 3.5

Set $- 4 \cos (x) + 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $- 4 \cos (x) + 1$ equal to $0$ .

$- 4 \cos (x) + 1 = 0$

Step 3.5.2

Solve $- 4 \cos (x) + 1 = 0$ for $x$ .

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Step 3.5.2.1

Subtract $1$ from both sides of the equation.

$- 4 \cos (x) = - 1$

Step 3.5.2.2

Divide each term in $- 4 \cos (x) = - 1$ by $- 4$ and simplify.

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Step 3.5.2.2.1

Divide each term in $- 4 \cos (x) = - 1$ by $- 4$ .

$\frac{- 4 \cos (x)}{- 4} = \frac{- 1}{- 4}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 \cos (x)}{- 4} = \frac{- 1}{- 4}$

Step 3.5.2.2.2.1.2

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

$\cos (x) = \frac{- 1}{- 4}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Dividing two negative values results in a positive value.

$\cos (x) = \frac{1}{4}$

Step 3.5.2.3

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

$x = \arccos (\frac{1}{4})$

Step 3.5.2.4

Simplify the right side.

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Step 3.5.2.4.1

Evaluate $\arccos (\frac{1}{4})$ .

$x = 1.31811607$

Step 3.5.2.5

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.

$x = 2 (3.14159265) - 1.31811607$

Step 3.5.2.6

Solve for $x$ .

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Step 3.5.2.6.1

Remove parentheses.

$x = 2 (3.14159265) - 1.31811607$

Step 3.5.2.6.2

Simplify $2 (3.14159265) - 1.31811607$ .

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Step 3.5.2.6.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.31811607$

Step 3.5.2.6.2.2

Subtract $1.31811607$ from $6.2831853$ .

$x = 4.96506923$

Step 3.5.2.7

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

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Step 3.5.2.7.1

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

$\frac{2 π}{| b |}$

Step 3.5.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.5.2.7.3

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

$\frac{2 π}{1}$

Step 3.5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.5.2.8

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

$x = 1.31811607 + 2 π n, 4.96506923 + 2 π n$ , for any integer $n$

Step 3.6

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

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Step 3.6.1

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

$\cos (x) - 1 = 0$

Step 3.6.2

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

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Step 3.6.2.1

Add $1$ to both sides of the equation.

$\cos (x) = 1$

Step 3.6.2.2

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

$x = \arccos (1)$

Step 3.6.2.3

Simplify the right side.

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Step 3.6.2.3.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 3.6.2.4

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.

$x = 2 π - 0$

Step 3.6.2.5

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 3.6.2.6

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

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Step 3.6.2.6.1

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

$\frac{2 π}{| b |}$

Step 3.6.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 3.6.2.6.3

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

$\frac{2 π}{1}$

Step 3.6.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.6.2.7

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

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

Step 3.7

The final solution is all the values that make $(- 4 \cos (x) + 1) (\cos (x) - 1) = 0$ true.

$x = 1.31811607 + 2 π n, 4.96506923 + 2 π n, 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 4

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

$x = 1.31811607 + 2 π n, 4.96506923 + 2 π n, 2 π n$ , for any integer $n$

Step 5

Exclude the solutions that do not make $\sqrt{\cos (x)} = 2 \cos (x) - 1$ true.

$x = 2 π n$ , for any integer $n$