Trigonometry Examples

$(\frac{5}{8}, - \sqrt{\frac{39}{8}})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{5}{8}, - \sqrt{\frac{39}{8}})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{5}{8}, 0)$ , and $(\frac{5}{8}, - \sqrt{\frac{39}{8}})$ .

Opposite : $- \sqrt{\frac{39}{8}}$

Adjacent : $\frac{5}{8}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{5}{8}$ .

$\sqrt{\frac{5^{2}}{8^{2}} + {(- \sqrt{\frac{39}{8}})}^{2}}$

Step 2.2

Raise $5$ to the power of $2$ .

$\sqrt{\frac{25}{8^{2}} + {(- \sqrt{\frac{39}{8}})}^{2}}$

Step 2.3

Raise $8$ to the power of $2$ .

$\sqrt{\frac{25}{64} + {(- \sqrt{\frac{39}{8}})}^{2}}$

Step 2.4

Rewrite $\sqrt{\frac{39}{8}}$ as $\frac{\sqrt{39}}{\sqrt{8}}$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39}}{\sqrt{8}})}^{2}}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

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

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Step 2.5.1.1

Factor $4$ out of $8$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39}}{\sqrt{4 (2)}})}^{2}}$

Step 2.5.1.2

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39}}{\sqrt{2^{2} \cdot 2}})}^{2}}$

Step 2.5.2

Pull terms out from under the radical.

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39}}{2 \sqrt{2}})}^{2}}$

Step 2.6

Multiply $\frac{\sqrt{39}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{25}{64} + {(- (\frac{\sqrt{39}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))}^{2}}$

Step 2.7

Combine and simplify the denominator.

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Step 2.7.1

Multiply $\frac{\sqrt{39}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 \sqrt{2} \sqrt{2}})}^{2}}$

Step 2.7.2

Move $\sqrt{2}$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})})}^{2}}$

Step 2.7.3

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})})}^{2}}$

Step 2.7.4

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})})}^{2}}$

Step 2.7.5

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}})}^{2}}$

Step 2.7.6

Add $1$ and $1$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 {\sqrt{2}}^{2}})}^{2}}$

Step 2.7.7

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

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Step 2.7.7.1

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}})}^{2}}$

Step 2.7.7.2

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}})}^{2}}$

Step 2.7.7.3

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

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}^{2}}$

Step 2.7.7.4

Cancel the common factor of $2$ .

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Step 2.7.7.4.1

Cancel the common factor.

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}^{2}}$

Step 2.7.7.4.2

Rewrite the expression.

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 \cdot 2^{1}})}^{2}}$

Step 2.7.7.5

Evaluate the exponent.

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39} \sqrt{2}}{2 \cdot 2})}^{2}}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Combine using the product rule for radicals.

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{39 \cdot 2}}{2 \cdot 2})}^{2}}$

Step 2.8.2

Multiply $39$ by $2$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{78}}{2 \cdot 2})}^{2}}$

Step 2.9

Multiply $2$ by $2$ .

$\sqrt{\frac{25}{64} + {(- \frac{\sqrt{78}}{4})}^{2}}$

Step 2.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.10.1

Apply the product rule to $- \frac{\sqrt{78}}{4}$ .

$\sqrt{\frac{25}{64} + {(- 1)}^{2} {(\frac{\sqrt{78}}{4})}^{2}}$

Step 2.10.2

Apply the product rule to $\frac{\sqrt{78}}{4}$ .

$\sqrt{\frac{25}{64} + {(- 1)}^{2} \frac{{\sqrt{78}}^{2}}{4^{2}}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Raise $- 1$ to the power of $2$ .

$\sqrt{\frac{25}{64} + 1 \frac{{\sqrt{78}}^{2}}{4^{2}}}$

Step 2.11.2

Multiply $\frac{{\sqrt{78}}^{2}}{4^{2}}$ by $1$ .

$\sqrt{\frac{25}{64} + \frac{{\sqrt{78}}^{2}}{4^{2}}}$

Step 2.12

Rewrite ${\sqrt{78}}^{2}$ as $78$ .

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Step 2.12.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{78}$ as $78^{\frac{1}{2}}$ .

$\sqrt{\frac{25}{64} + \frac{{(78^{\frac{1}{2}})}^{2}}{4^{2}}}$

Step 2.12.2

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

$\sqrt{\frac{25}{64} + \frac{78^{\frac{1}{2} \cdot 2}}{4^{2}}}$

Step 2.12.3

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

$\sqrt{\frac{25}{64} + \frac{78^{\frac{2}{2}}}{4^{2}}}$

Step 2.12.4

Cancel the common factor of $2$ .

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Step 2.12.4.1

Cancel the common factor.

$\sqrt{\frac{25}{64} + \frac{78^{\frac{2}{2}}}{4^{2}}}$

Step 2.12.4.2

Rewrite the expression.

$\sqrt{\frac{25}{64} + \frac{78^{1}}{4^{2}}}$

Step 2.12.5

Evaluate the exponent.

$\sqrt{\frac{25}{64} + \frac{78}{4^{2}}}$

Step 2.13

Raise $4$ to the power of $2$ .

$\sqrt{\frac{25}{64} + \frac{78}{16}}$

Step 2.14

Cancel the common factor of $78$ and $16$ .

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Step 2.14.1

Factor $2$ out of $78$ .

$\sqrt{\frac{25}{64} + \frac{2 (39)}{16}}$

Step 2.14.2

Cancel the common factors.

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Step 2.14.2.1

Factor $2$ out of $16$ .

$\sqrt{\frac{25}{64} + \frac{2 \cdot 39}{2 \cdot 8}}$

Step 2.14.2.2

Cancel the common factor.

$\sqrt{\frac{25}{64} + \frac{2 \cdot 39}{2 \cdot 8}}$

Step 2.14.2.3

Rewrite the expression.

$\sqrt{\frac{25}{64} + \frac{39}{8}}$

Step 2.15

To write $\frac{39}{8}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\sqrt{\frac{25}{64} + \frac{39}{8} \cdot \frac{8}{8}}$

Step 2.16

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.16.1

Multiply $\frac{39}{8}$ by $\frac{8}{8}$ .

$\sqrt{\frac{25}{64} + \frac{39 \cdot 8}{8 \cdot 8}}$

Step 2.16.2

Multiply $8$ by $8$ .

$\sqrt{\frac{25}{64} + \frac{39 \cdot 8}{64}}$

Step 2.17

Combine the numerators over the common denominator.

$\sqrt{\frac{25 + 39 \cdot 8}{64}}$

Step 2.18

Simplify the numerator.

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Step 2.18.1

Multiply $39$ by $8$ .

$\sqrt{\frac{25 + 312}{64}}$

Step 2.18.2

Add $25$ and $312$ .

$\sqrt{\frac{337}{64}}$

Step 2.19

Rewrite $\sqrt{\frac{337}{64}}$ as $\frac{\sqrt{337}}{\sqrt{64}}$ .

$\frac{\sqrt{337}}{\sqrt{64}}$

Step 2.20

Simplify the denominator.

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Step 2.20.1

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{337}}{\sqrt{8^{2}}}$

Step 2.20.2

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

$\frac{\sqrt{337}}{8}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{5}{8}}{\frac{\sqrt{337}}{8}}$ .

$\frac{\frac{5}{8}}{\frac{\sqrt{337}}{8}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = \frac{5}{8} \cdot \frac{8}{\sqrt{337}}$

Step 4.2

Cancel the common factor of $8$ .

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Step 4.2.1

Cancel the common factor.

$\cos (θ) = \frac{5}{8} \cdot \frac{8}{\sqrt{337}}$

Step 4.2.2

Rewrite the expression.

$\cos (θ) = 5 (\frac{1}{\sqrt{337}})$

Step 4.3

Combine $5$ and $\frac{1}{\sqrt{337}}$ .

$\cos (θ) = \frac{5}{\sqrt{337}}$

Step 4.4

Multiply $\frac{5}{\sqrt{337}}$ by $\frac{\sqrt{337}}{\sqrt{337}}$ .

$\cos (θ) = \frac{5}{\sqrt{337}} \cdot \frac{\sqrt{337}}{\sqrt{337}}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{5}{\sqrt{337}}$ by $\frac{\sqrt{337}}{\sqrt{337}}$ .

$\cos (θ) = \frac{5 \sqrt{337}}{\sqrt{337} \sqrt{337}}$

Step 4.5.2

Raise $\sqrt{337}$ to the power of $1$ .

$\cos (θ) = \frac{5 \sqrt{337}}{\sqrt{337} \sqrt{337}}$

Step 4.5.3

Raise $\sqrt{337}$ to the power of $1$ .

$\cos (θ) = \frac{5 \sqrt{337}}{\sqrt{337} \sqrt{337}}$

Step 4.5.4

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

$\cos (θ) = \frac{5 \sqrt{337}}{{\sqrt{337}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\cos (θ) = \frac{5 \sqrt{337}}{{\sqrt{337}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{337}}^{2}$ as $337$ .

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{337}$ as $337^{\frac{1}{2}}$ .

$\cos (θ) = \frac{5 \sqrt{337}}{{(337^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\cos (θ) = \frac{5 \sqrt{337}}{337^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\cos (θ) = \frac{5 \sqrt{337}}{337^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{5 \sqrt{337}}{337^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{5 \sqrt{337}}{337}$

Step 4.5.6.5

Evaluate the exponent.

$\cos (θ) = \frac{5 \sqrt{337}}{337}$

Step 5

Approximate the result.

$\cos (θ) = \frac{5 \sqrt{337}}{337} \approx 0.27236735$