Algebra Examples

$3 = \tan (2 x - π)$ in $(\frac{π}{2}, \frac{3 π}{4})$

Step 1

Rewrite the equation as $\tan (2 x - π) = 3$ .

$\tan (2 x - π) = 3$

Step 2

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

$2 x - π = \arctan (3)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arctan (3)$ .

$2 x - π = 1.24904577$

Step 4

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

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Step 4.1

Add $π$ to both sides of the equation.

$2 x = 1.24904577 + π$

Step 4.2

Replace with decimal approximation.

$2 x = 1.24904577 + 3.14159265$

Step 4.3

Add $1.24904577$ and $3.14159265$ .

$2 x = 4.39063842$

Step 5

Divide each term in $2 x = 4.39063842$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = 4.39063842$ by $2$ .

$\frac{2 x}{2} = \frac{4.39063842}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.39063842}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.39063842}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $4.39063842$ by $2$ .

$x = 2.19531921$

Step 6

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.

$2 x - π = (3.14159265) + 1.24904577$

Step 7

Solve for $x$ .

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Step 7.1

Add $3.14159265$ and $1.24904577$ .

$2 x - π = 4.39063842$

Step 7.2

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

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Step 7.2.1

Add $π$ to both sides of the equation.

$2 x = 4.39063842 + π$

Step 7.2.2

Replace with decimal approximation.

$2 x = 4.39063842 + 3.14159265$

Step 7.2.3

Add $4.39063842$ and $3.14159265$ .

$2 x = 7.53223107$

Step 7.3

Divide each term in $2 x = 7.53223107$ by $2$ and simplify.

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Step 7.3.1

Divide each term in $2 x = 7.53223107$ by $2$ .

$\frac{2 x}{2} = \frac{7.53223107}{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7.53223107}{2}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{7.53223107}{2}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Divide $7.53223107$ by $2$ .

$x = 3.76611553$

Step 8

Find the period of $\tan (2 x - π)$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| 2 |}$

Step 8.3

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

$\frac{π}{2}$

Step 9

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

$x = 2.19531921 + \frac{π n}{2}, 3.76611553 + \frac{π n}{2}$ , for any integer $n$

Step 10

Find the values of $n$ that produce a value within the interval $(\frac{π}{2}, \frac{3 π}{4})$ .

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Step 10.1

Plug in $- 1$ for $n$ and simplify to see if the solution is contained in $(\frac{π}{2}, \frac{3 π}{4})$ .

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Step 10.1.1

Plug in $- 1$ for $n$ .

$3.76611553 + \frac{π (- 1)}{2}$

Step 10.1.2

Simplify.

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Step 10.1.2.1

Simplify each term.

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Step 10.1.2.1.1

Move $- 1$ to the left of $π$ .

$3.76611553 + \frac{- 1 \cdot π}{2}$

Step 10.1.2.1.2

Move the negative in front of the fraction.

$3.76611553 - \frac{π}{2}$

Step 10.1.2.2

Subtract $\frac{π}{2}$ from $3.76611553$ .

$2.19531921$

Step 10.1.3

The interval $(\frac{π}{2}, \frac{3 π}{4})$ contains $2.19531921$ .

$x = 2.19531921$

Step 10.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(\frac{π}{2}, \frac{3 π}{4})$ .

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Step 10.2.1

Plug in $0$ for $n$ .

$2.19531921 + \frac{π (0)}{2}$

Step 10.2.2

Simplify.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

Cancel the common factor of $0$ and $2$ .

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Step 10.2.2.1.1.1

Factor $2$ out of $π (0)$ .

$2.19531921 + \frac{2 (π \cdot (0))}{2}$

Step 10.2.2.1.1.2

Cancel the common factors.

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Step 10.2.2.1.1.2.1

Factor $2$ out of $2$ .

$2.19531921 + \frac{2 (π \cdot (0))}{2 (1)}$

Step 10.2.2.1.1.2.2

Cancel the common factor.

$2.19531921 + \frac{2 (π \cdot (0))}{2 \cdot 1}$

Step 10.2.2.1.1.2.3

Rewrite the expression.

$2.19531921 + \frac{π \cdot (0)}{1}$

Step 10.2.2.1.1.2.4

Divide $π \cdot (0)$ by $1$ .

$2.19531921 + π \cdot (0)$

Step 10.2.2.1.2

Multiply $π$ by $0$ .

$2.19531921 + 0$

Step 10.2.2.2

Add $2.19531921$ and $0$ .

$2.19531921$

Step 10.2.3

The interval $(\frac{π}{2}, \frac{3 π}{4})$ contains $2.19531921$ .

$x = 2.19531921, 2.19531921$