Trigonometry Examples

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

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\sqrt{7}, \sqrt{5})$ , draw the triangle between the three points $(0, 0)$ , $(\sqrt{7}, 0)$ , and $(\sqrt{7}, \sqrt{5})$ .

Opposite : $\sqrt{5}$

Adjacent : $\sqrt{7}$

Step 2

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

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

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

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

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

Step 3.2

Combine and simplify the denominator.

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Step 3.2.1

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

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

Step 3.2.2

Raise $\sqrt{7}$ to the power of $1$ .

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

Step 3.2.3

Raise $\sqrt{7}$ to the power of $1$ .

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

Step 3.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.5

Add $1$ and $1$ .

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

Step 3.2.6

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

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Step 3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

Step 3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.2.6.4

Cancel the common factor of $2$ .

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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.

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