Trigonometry Examples

$y = \frac{\sqrt{5 x - 2}}{x^{2} - 16}$

Step 1

Find the $y$ value at $x = 1$ .

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Step 1.1

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

$f (1) = \frac{\sqrt{5 (1) - 2}}{((1) + 4) ((1) - 4)}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Multiply $5$ by $1$ .

$f (1) = \frac{\sqrt{5 - 2}}{(1 + 4) (1 - 4)}$

Step 1.2.1.2

Subtract $2$ from $5$ .

$f (1) = \frac{\sqrt{3}}{(1 + 4) (1 - 4)}$

Step 1.2.2

Simplify the denominator.

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Step 1.2.2.1

Add $1$ and $4$ .

$f (1) = \frac{\sqrt{3}}{5 (1 - 4)}$

Step 1.2.2.2

Subtract $4$ from $1$ .

$f (1) = \frac{\sqrt{3}}{5 \cdot - 3}$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $5$ by $- 3$ .

$f (1) = \frac{\sqrt{3}}{- 15}$

Step 1.2.3.2

Move the negative in front of the fraction.

$f (1) = - \frac{\sqrt{3}}{15}$

Step 1.2.4

The final answer is $- \frac{\sqrt{3}}{15}$ .

$- \frac{\sqrt{3}}{15}$

Step 1.3

The $y$ value at $x = 1$ is $- \frac{\sqrt{3}}{15}$ .

$y = - \frac{\sqrt{3}}{15}$

Step 2

Find the $y$ value at $x = 2$ .

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Step 2.1

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

$f (2) = \frac{\sqrt{5 (2) - 2}}{((2) + 4) ((2) - 4)}$

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

Multiply $5$ by $2$ .

$f (2) = \frac{\sqrt{10 - 2}}{(2 + 4) (2 - 4)}$

Step 2.2.1.2

Subtract $2$ from $10$ .

$f (2) = \frac{\sqrt{8}}{(2 + 4) (2 - 4)}$

Step 2.2.1.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.2.1.3.1

Factor $4$ out of $8$ .

$f (2) = \frac{\sqrt{4 (2)}}{(2 + 4) (2 - 4)}$

Step 2.2.1.3.2

Rewrite $4$ as $2^{2}$ .

$f (2) = \frac{\sqrt{2^{2} \cdot 2}}{(2 + 4) (2 - 4)}$

Step 2.2.1.4

Pull terms out from under the radical.

$f (2) = \frac{2 \sqrt{2}}{(2 + 4) (2 - 4)}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Add $2$ and $4$ .

$f (2) = \frac{2 \sqrt{2}}{6 (2 - 4)}$

Step 2.2.2.2

Subtract $4$ from $2$ .

$f (2) = \frac{2 \sqrt{2}}{6 \cdot - 2}$

Step 2.2.3

Reduce the expression by cancelling the common factors.

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Step 2.2.3.1

Multiply $6$ by $- 2$ .

$f (2) = \frac{2 \sqrt{2}}{- 12}$

Step 2.2.3.2

Cancel the common factor of $2$ and $- 12$ .

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Step 2.2.3.2.1

Factor $2$ out of $2 \sqrt{2}$ .

$f (2) = \frac{2 (\sqrt{2})}{- 12}$

Step 2.2.3.2.2

Cancel the common factors.

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Step 2.2.3.2.2.1

Factor $2$ out of $- 12$ .

$f (2) = \frac{2 \sqrt{2}}{2 \cdot - 6}$

Step 2.2.3.2.2.2

Cancel the common factor.

$f (2) = \frac{2 \sqrt{2}}{2 \cdot - 6}$

Step 2.2.3.2.2.3

Rewrite the expression.

$f (2) = \frac{\sqrt{2}}{- 6}$

Step 2.2.3.3

Move the negative in front of the fraction.

$f (2) = - \frac{\sqrt{2}}{6}$

Step 2.2.4

The final answer is $- \frac{\sqrt{2}}{6}$ .

$- \frac{\sqrt{2}}{6}$

Step 2.3

The $y$ value at $x = 2$ is $- \frac{\sqrt{2}}{6}$ .

$y = - \frac{\sqrt{2}}{6}$

Step 3

Find the $y$ value at $x = 3$ .

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{\sqrt{5 (3) - 2}}{((3) + 4) ((3) - 4)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Multiply $5$ by $3$ .

$f (3) = \frac{\sqrt{15 - 2}}{(3 + 4) (3 - 4)}$

Step 3.2.1.2

Subtract $2$ from $15$ .

$f (3) = \frac{\sqrt{13}}{(3 + 4) (3 - 4)}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Add $3$ and $4$ .

$f (3) = \frac{\sqrt{13}}{7 (3 - 4)}$

Step 3.2.2.2

Subtract $4$ from $3$ .

$f (3) = \frac{\sqrt{13}}{7 \cdot - 1}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $7$ by $- 1$ .

$f (3) = \frac{\sqrt{13}}{- 7}$

Step 3.2.3.2

Move the negative in front of the fraction.

$f (3) = - \frac{\sqrt{13}}{7}$

Step 3.2.4

The final answer is $- \frac{\sqrt{13}}{7}$ .

$- \frac{\sqrt{13}}{7}$

Step 3.3

The $y$ value at $x = 3$ is $- \frac{\sqrt{13}}{7}$ .

$y = - \frac{\sqrt{13}}{7}$

Step 4

Find the $y$ value at $x = 5$ .

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Step 4.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{\sqrt{5 (5) - 2}}{((5) + 4) ((5) - 4)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Multiply $5$ by $5$ .

$f (5) = \frac{\sqrt{25 - 2}}{(5 + 4) (5 - 4)}$

Step 4.2.1.2

Subtract $2$ from $25$ .

$f (5) = \frac{\sqrt{23}}{(5 + 4) (5 - 4)}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Add $5$ and $4$ .

$f (5) = \frac{\sqrt{23}}{9 (5 - 4)}$

Step 4.2.2.2

Subtract $4$ from $5$ .

$f (5) = \frac{\sqrt{23}}{9 \cdot 1}$

Step 4.2.3

Multiply $9$ by $1$ .

$f (5) = \frac{\sqrt{23}}{9}$

Step 4.2.4

The final answer is $\frac{\sqrt{23}}{9}$ .

$\frac{\sqrt{23}}{9}$

Step 4.3

The $y$ value at $x = 5$ is $\frac{\sqrt{23}}{9}$ .

$y = \frac{\sqrt{23}}{9}$

Step 5

Find the $y$ value at $x = 6$ .

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Step 5.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{\sqrt{5 (6) - 2}}{((6) + 4) ((6) - 4)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $5$ by $6$ .

$f (6) = \frac{\sqrt{30 - 2}}{(6 + 4) (6 - 4)}$

Step 5.2.1.2

Subtract $2$ from $30$ .

$f (6) = \frac{\sqrt{28}}{(6 + 4) (6 - 4)}$

Step 5.2.1.3

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 5.2.1.3.1

Factor $4$ out of $28$ .

$f (6) = \frac{\sqrt{4 (7)}}{(6 + 4) (6 - 4)}$

Step 5.2.1.3.2

Rewrite $4$ as $2^{2}$ .

$f (6) = \frac{\sqrt{2^{2} \cdot 7}}{(6 + 4) (6 - 4)}$

Step 5.2.1.4

Pull terms out from under the radical.

$f (6) = \frac{2 \sqrt{7}}{(6 + 4) (6 - 4)}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Add $6$ and $4$ .

$f (6) = \frac{2 \sqrt{7}}{10 (6 - 4)}$

Step 5.2.2.2

Subtract $4$ from $6$ .

$f (6) = \frac{2 \sqrt{7}}{10 \cdot 2}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Multiply $10$ by $2$ .

$f (6) = \frac{2 \sqrt{7}}{20}$

Step 5.2.3.2

Cancel the common factor of $2$ and $20$ .

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Step 5.2.3.2.1

Factor $2$ out of $2 \sqrt{7}$ .

$f (6) = \frac{2 (\sqrt{7})}{20}$

Step 5.2.3.2.2

Cancel the common factors.

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Step 5.2.3.2.2.1

Factor $2$ out of $20$ .

$f (6) = \frac{2 \sqrt{7}}{2 \cdot 10}$

Step 5.2.3.2.2.2

Cancel the common factor.

$f (6) = \frac{2 \sqrt{7}}{2 \cdot 10}$

Step 5.2.3.2.2.3

Rewrite the expression.

$f (6) = \frac{\sqrt{7}}{10}$

Step 5.2.4

The final answer is $\frac{\sqrt{7}}{10}$ .

$\frac{\sqrt{7}}{10}$

Step 5.3

The $y$ value at $x = 6$ is $\frac{\sqrt{7}}{10}$ .

$y = \frac{\sqrt{7}}{10}$

Step 6

Find the $y$ value at $x = 7$ .

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Step 6.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{\sqrt{5 (7) - 2}}{((7) + 4) ((7) - 4)}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $5$ by $7$ .

$f (7) = \frac{\sqrt{35 - 2}}{(7 + 4) (7 - 4)}$

Step 6.2.1.2

Subtract $2$ from $35$ .

$f (7) = \frac{\sqrt{33}}{(7 + 4) (7 - 4)}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $7$ and $4$ .

$f (7) = \frac{\sqrt{33}}{11 (7 - 4)}$

Step 6.2.2.2

Subtract $4$ from $7$ .

$f (7) = \frac{\sqrt{33}}{11 \cdot 3}$

Step 6.2.3

Multiply $11$ by $3$ .

$f (7) = \frac{\sqrt{33}}{33}$

Step 6.2.4

The final answer is $\frac{\sqrt{33}}{33}$ .

$\frac{\sqrt{33}}{33}$

Step 6.3

The $y$ value at $x = 7$ is $\frac{\sqrt{33}}{33}$ .

$y = \frac{\sqrt{33}}{33}$

Step 7

List the points to graph.

$(1, - 0.11547005), (2, - 0.23570226), (3, - 0.51507875), (5, 0.53287016), (6, 0.26457513), (7, 0.17407765)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ 1 & - 0.115 \\ 2 & - 0.236 \\ 3 & - 0.515 \\ 5 & 0.533 \\ 6 & 0.265 \\ 7 & 0.174 \end{array}$

Step 9