Trigonometry Examples

$(\sqrt{3}, \sqrt{5})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\sqrt{3}, \sqrt{5})$ , draw the triangle between the three points $(0, 0)$ , $(\sqrt{3}, 0)$ , and $(\sqrt{3}, \sqrt{5})$ .

Opposite : $\sqrt{5}$

Adjacent : $\sqrt{3}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{{(3^{\frac{1}{2}})}^{2} + {(\sqrt{5})}^{2}}$

Step 2.1.2

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

$\sqrt{3^{\frac{1}{2} \cdot 2} + {(\sqrt{5})}^{2}}$

Step 2.1.3

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

$\sqrt{3^{\frac{2}{2}} + {(\sqrt{5})}^{2}}$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\sqrt{3^{\frac{2}{2}} + {(\sqrt{5})}^{2}}$

Step 2.1.4.2

Rewrite the expression.

$\sqrt{3^{1} + {(\sqrt{5})}^{2}}$

Step 2.1.5

Evaluate the exponent.

$\sqrt{3 + {(\sqrt{5})}^{2}}$

Step 2.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{3 + {(5^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{3 + 5^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{3 + 5^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{3 + 5^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{3 + 5^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{3 + 5}$

Step 2.3

Add $3$ and $5$ .

$\sqrt{8}$

Step 2.4

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

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Step 2.4.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 2.4.2

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

$\sqrt{2^{2} \cdot 2}$

Step 2.5

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\sqrt{3}}{2 \sqrt{2}}$ .

$\frac{\sqrt{3}}{2 \sqrt{2}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply $\frac{\sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (θ) = \frac{\sqrt{3}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{\sqrt{3}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 4.2.2

Move $\sqrt{2}$ .

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.2.4

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.2.5

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

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 4.2.6

Add $1$ and $1$ .

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 4.2.7

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

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Step 4.2.7.1

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

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 4.2.7.2

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

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 4.2.7.3

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

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 4.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.7.4.1

Cancel the common factor.

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 4.2.7.4.2

Rewrite the expression.

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 4.2.7.5

Evaluate the exponent.

$\cos (θ) = \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Combine using the product rule for radicals.

$\cos (θ) = \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}$

Step 4.3.2

Multiply $3$ by $2$ .

$\cos (θ) = \frac{\sqrt{6}}{2 \cdot 2}$

Step 4.4

Multiply $2$ by $2$ .

$\cos (θ) = \frac{\sqrt{6}}{4}$

Step 5

Approximate the result.

$\cos (θ) = \frac{\sqrt{6}}{4} \approx 0.61237243$