Trigonometry Examples

(\sqrt{7}, \sqrt{5})

$(\sqrt{7}, \sqrt{5})$

Step 1

To find the

tan (θ)

$\tan (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(\sqrt{7}, \sqrt{5})

$(\sqrt{7}, \sqrt{5})$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(\sqrt{7}, 0)

$(\sqrt{7}, 0)$ , and

(\sqrt{7}, \sqrt{5})

$(\sqrt{7}, \sqrt{5})$ .

Opposite :

\sqrt{5}

$\sqrt{5}$

Adjacent :

\sqrt{7}

$\sqrt{7}$

Step 2

tan (θ) = \frac{Opposite}{Adjacent}

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore

tan (θ) = \frac{\sqrt{5}}{\sqrt{7}}

$\tan (θ) = \frac{\sqrt{5}}{\sqrt{7}}$ .

\frac{\sqrt{5}}{\sqrt{7}}

$\frac{\sqrt{5}}{\sqrt{7}}$

Step 3

Simplify

tan (θ)

$\tan (θ)$ .

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Step 3.1

Multiply

\frac{\sqrt{5}}{\sqrt{7}}

$\frac{\sqrt{5}}{\sqrt{7}}$ by

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

tan (θ) = \frac{\sqrt{5}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}

$\tan (θ) = \frac{\sqrt{5}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 3.2

Combine and simplify the denominator.

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Step 3.2.1

Multiply

\frac{\sqrt{5}}{\sqrt{7}}

$\frac{\sqrt{5}}{\sqrt{7}}$ by

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.2.2

Raise

\sqrt{7}

$\sqrt{7}$ to the power of

1

$1$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.2.3

Raise

\sqrt{7}

$\sqrt{7}$ to the power of

1

$1$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.2.5

Add

1

$1$ and

1

$1$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{2}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.2.6

Rewrite

{\sqrt{7}}^{2}

${\sqrt{7}}^{2}$ as

7

$7$ .

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Step 3.2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{7}

$\sqrt{7}$ as

7^{\frac{1}{2}}

$7^{\frac{1}{2}}$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.2.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}}

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.2.6.4

Cancel the common factor of

2

$2$ .

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Step 3.2.6.4.1

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.2.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7}$

Step 3.2.6.5

Evaluate the exponent.

$\tan (θ) = \frac{\sqrt{5} \sqrt{7}}{7}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Combine using the product rule for radicals.

$\tan (θ) = \frac{\sqrt{5 \cdot 7}}{7}$

Step 3.3.2

Multiply

$5$ by

$7$ .

$\tan (θ) = \frac{\sqrt{35}}{7}$

Step 4

Approximate the result.

$\tan (θ) = \frac{\sqrt{35}}{7} \approx 0.84515425$