Algebra Examples

$5 \cos (2 x + 3) = \sin (2 x + 3)$

Step 1

Divide each term in the equation by $\cos (2 x + 3)$ .

$\frac{5 \cos (2 x + 3)}{\cos (2 x + 3)} = \frac{\sin (2 x + 3)}{\cos (2 x + 3)}$

Step 2

Cancel the common factor of $\cos (2 x + 3)$ .

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Step 2.1

Cancel the common factor.

$\frac{5 \cos (2 x + 3)}{\cos (2 x + 3)} = \frac{\sin (2 x + 3)}{\cos (2 x + 3)}$

Step 2.2

Divide $5$ by $1$ .

$5 = \frac{\sin (2 x + 3)}{\cos (2 x + 3)}$

Step 3

Convert from $\frac{\sin (2 x + 3)}{\cos (2 x + 3)}$ to $\tan (2 x + 3)$ .

$5 = \tan (2 x + 3)$

Step 4

Rewrite the equation as $\tan (2 x + 3) = 5$ .

$\tan (2 x + 3) = 5$

Step 5

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

$2 x + 3 = \arctan (5)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arctan (5)$ .

$2 x + 3 = 1.37340076$

Step 7

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1

Subtract $3$ from both sides of the equation.

$2 x = 1.37340076 - 3$

Step 7.2

Subtract $3$ from $1.37340076$ .

$2 x = - 1.62659923$

Step 8

Divide each term in $2 x = - 1.62659923$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 x = - 1.62659923$ by $2$ .

$\frac{2 x}{2} = \frac{- 1.62659923}{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1.62659923}{2}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1.62659923}{2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $- 1.62659923$ by $2$ .

$x = - 0.81329961$

Step 9

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$2 x + 3 = (3.14159265) + 1.37340076$

Step 10

Solve for $x$ .

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Step 10.1

Add $3.14159265$ and $1.37340076$ .

$2 x + 3 = 4.51499342$

Step 10.2

Move all terms not containing $x$ to the right side of the equation.

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Step 10.2.1

Subtract $3$ from both sides of the equation.

$2 x = 4.51499342 - 3$

Step 10.2.2

Subtract $3$ from $4.51499342$ .

$2 x = 1.51499342$

Step 10.3

Divide each term in $2 x = 1.51499342$ by $2$ and simplify.

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Step 10.3.1

Divide each term in $2 x = 1.51499342$ by $2$ .

$\frac{2 x}{2} = \frac{1.51499342}{2}$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $2$ .

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Step 10.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.51499342}{2}$

Step 10.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.51499342}{2}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Divide $1.51499342$ by $2$ .

$x = 0.75749671$

Step 11

Find the period of $\tan (2 x + 3)$ .

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Step 11.1

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

$\frac{π}{| b |}$

Step 11.2

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

$\frac{π}{| 2 |}$

Step 11.3

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

$\frac{π}{2}$

Step 12

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Step 12.1

Add $\frac{π}{2}$ to $- 0.81329961$ to find the positive angle.

$- 0.81329961 + \frac{π}{2}$

Step 12.2

Subtract $0.81329961$ from $\frac{π}{2}$ .

$0.75749671$

Step 12.3

List the new angles.

$x = 0.75749671$

Step 13

The period of the $\tan (2 x + 3)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = 0.75749671 + \frac{π n}{2}, 0.75749671 + \frac{π n}{2}$ , for any integer $n$

Step 14

Consolidate $0.75749671 + \frac{π n}{2}$ and $0.75749671 + \frac{π n}{2}$ to $0.75749671 + \frac{π n}{2}$ .

$x = 0.75749671 + \frac{π n}{2}$ , for any integer $n$