Trigonometry Examples

$2 \arctan (\frac{2 (\sqrt{b})}{b + 1})$

Step 1

Find the domain for $y = 2 \arctan (\frac{2 (\sqrt{b})}{b + 1})$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 1.2

Set the denominator in $\frac{2 \sqrt{x}}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Step 1.3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = 2 \arctan (\frac{2 \sqrt{x}}{x + 1})$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 \arctan (\frac{2 \sqrt{0}}{(0) + 1})$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

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

$f (0) = 2 \arctan (\frac{2 \sqrt{0^{2}}}{0 + 1})$

Step 2.2.1.2

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

$f (0) = 2 \arctan (\frac{2 \cdot 0}{0 + 1})$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Add $0$ and $1$ .

$f (0) = 2 \arctan (\frac{2 \cdot 0}{1})$

Step 2.2.2.2

Multiply $2$ by $0$ .

$f (0) = 2 \arctan (\frac{0}{1})$

Step 2.2.2.3

Divide $0$ by $1$ .

$f (0) = 2 \arctan (0)$

Step 2.2.3

The exact value of $\arctan (0)$ is $0$ .

$f (0) = 2 \cdot 0$

Step 2.2.4

Multiply $2$ by $0$ .

$f (0) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(0, 0)$ .

$(0, 0)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $1$ into $f (x) = 2 \arctan (\frac{2 \sqrt{x}}{x + 1})$ . In this case, the point is $(1, \frac{π}{2})$ .

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Step 4.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 \arctan (\frac{2 \sqrt{1}}{(1) + 1})$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Any root of $1$ is $1$ .

$f (1) = 2 \arctan (\frac{2 \cdot 1}{1 + 1})$

Step 4.1.2.2

Add $1$ and $1$ .

$f (1) = 2 \arctan (\frac{2 \cdot 1}{2})$

Step 4.1.2.3

Multiply $2$ by $1$ .

$f (1) = 2 \arctan (\frac{2}{2})$

Step 4.1.2.4

Divide $2$ by $2$ .

$f (1) = 2 \arctan (1)$

Step 4.1.2.5

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$f (1) = 2 (\frac{π}{4})$

Step 4.1.2.6

Cancel the common factor of $2$ .

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Step 4.1.2.6.1

Factor $2$ out of $4$ .

$f (1) = 2 (\frac{π}{2 (2)})$

Step 4.1.2.6.2

Cancel the common factor.

$f (1) = 2 (\frac{π}{2 \cdot 2})$

Step 4.1.2.6.3

Rewrite the expression.

$f (1) = \frac{π}{2}$

Step 4.1.2.7

The final answer is $\frac{π}{2}$ .

$y = \frac{π}{2}$

Step 4.2

Substitute the $x$ value $2$ into $f (x) = 2 \arctan (\frac{2 \sqrt{x}}{x + 1})$ . In this case, the point is $(2, 1.51193882)$ .

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Step 4.2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 2 \arctan (\frac{2 \sqrt{2}}{(2) + 1})$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Add $2$ and $1$ .

$f (2) = 2 \arctan (\frac{2 \sqrt{2}}{3})$

Step 4.2.2.2

Evaluate $\arctan (\frac{2 \sqrt{2}}{3})$ .

$f (2) = 2 \cdot 0.75596941$

Step 4.2.2.3

Multiply $2$ by $0.75596941$ .

$f (2) = 1.51193882$

Step 4.2.2.4

The final answer is $1.51193882$ .

$y = 1.51193882$

Step 4.3

The square root can be graphed using the points around the vertex $(0, 0), (1, 1.57), (2, 1.51)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1.57 \\ 2 & 1.51 \end{array}$

Step 5