Algebra Examples

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

Step 1

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (- \frac{2}{3})$ .

$2 A = 2.30052398$

Step 4

Divide each term in $2 A = 2.30052398$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 A = 2.30052398$ by $2$ .

$\frac{2 A}{2} = \frac{2.30052398}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 A}{2} = \frac{2.30052398}{2}$

Step 4.2.1.2

Divide $A$ by $1$ .

$A = \frac{2.30052398}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $2.30052398$ by $2$ .

$A = 1.15026199$

Step 5

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.

$2 A = 2 (3.14159265) - 2.30052398$

Step 6

Solve for $A$ .

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $2$ by $3.14159265$ .

$2 A = 6.2831853 - 2.30052398$

Step 6.1.2

Subtract $2.30052398$ from $6.2831853$ .

$2 A = 3.98266132$

Step 6.2

Divide each term in $2 A = 3.98266132$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 A = 3.98266132$ by $2$ .

$\frac{2 A}{2} = \frac{3.98266132}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 A}{2} = \frac{3.98266132}{2}$

Step 6.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{3.98266132}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $3.98266132$ by $2$ .

$A = 1.99133066$

Step 7

Find the period of $\cos (2 A)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 2 |}$

Step 7.3

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

$\frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

$A = 1.15026199 + π n, 1.99133066 + π n$ , for any integer $n$