Trigonometry Examples

$(- \frac{\sqrt{3}}{\sqrt{2}}, \frac{1}{2})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{3}}{\sqrt{2}}, \frac{1}{2})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{3}}{\sqrt{2}}, 0)$ , and $(- \frac{\sqrt{3}}{\sqrt{2}}, \frac{1}{2})$ .

Opposite : $\frac{1}{2}$

Adjacent : $- \frac{\sqrt{3}}{\sqrt{2}}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.2.6

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

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Step 2.2.6.1

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

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

Step 2.2.6.2

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

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

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

Step 2.2.6.4.2

Rewrite the expression.

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

Step 2.2.6.5

Evaluate the exponent.

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

Combine using the product rule for radicals.

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

Step 2.3.2

Multiply $3$ by $2$ .

$\sqrt{{(- \frac{\sqrt{6}}{2})}^{2} + {(\frac{1}{2})}^{2}}$

Step 2.4

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

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Step 2.4.1

Apply the product rule to $- \frac{\sqrt{6}}{2}$ .

$\sqrt{{(- 1)}^{2} {(\frac{\sqrt{6}}{2})}^{2} + {(\frac{1}{2})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

$\sqrt{{(- 1)}^{2} \frac{{\sqrt{6}}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

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

$\sqrt{1 \frac{{\sqrt{6}}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.5.2

Multiply $\frac{{\sqrt{6}}^{2}}{2^{2}}$ by $1$ .

$\sqrt{\frac{{\sqrt{6}}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{\frac{{(6^{\frac{1}{2}})}^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.6.2

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

$\sqrt{\frac{6^{\frac{1}{2} \cdot 2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.6.3

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

$\sqrt{\frac{6^{\frac{2}{2}}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{6^{\frac{2}{2}}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{6^{1}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{6}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.7

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

$\sqrt{\frac{6}{4} + {(\frac{1}{2})}^{2}}$

Step 2.8

Cancel the common factor of $6$ and $4$ .

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Step 2.8.1

Factor $2$ out of $6$ .

$\sqrt{\frac{2 (3)}{4} + {(\frac{1}{2})}^{2}}$

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $2$ out of $4$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.9

Simplify the expression.

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Step 2.9.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.9.2

One to any power is one.

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

Step 2.9.3

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

$\sqrt{\frac{3}{2} + \frac{1}{4}}$

Step 2.10

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{3}{2} \cdot \frac{2}{2} + \frac{1}{4}}$

Step 2.11

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.1

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

$\sqrt{\frac{3 \cdot 2}{2 \cdot 2} + \frac{1}{4}}$

Step 2.11.2

Multiply $2$ by $2$ .

$\sqrt{\frac{3 \cdot 2}{4} + \frac{1}{4}}$

Step 2.12

Combine the numerators over the common denominator.

$\sqrt{\frac{3 \cdot 2 + 1}{4}}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Multiply $3$ by $2$ .

$\sqrt{\frac{6 + 1}{4}}$

Step 2.13.2

Add $6$ and $1$ .

$\sqrt{\frac{7}{4}}$

Step 2.14

Rewrite $\sqrt{\frac{7}{4}}$ as $\frac{\sqrt{7}}{\sqrt{4}}$ .

$\frac{\sqrt{7}}{\sqrt{4}}$

Step 2.15

Simplify the denominator.

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Step 2.15.1

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

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

Step 2.15.2

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

$\frac{\sqrt{7}}{2}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{\sqrt{3}}{\sqrt{2}}}{\frac{\sqrt{7}}{2}}$ .

$\frac{- \frac{\sqrt{3}}{\sqrt{2}}}{\frac{\sqrt{7}}{2}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.2

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

$\cos (θ) = - \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.2

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.3

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.4

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.6

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

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Step 4.3.6.1

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.6.2

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.6.3

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

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Move the leading negative in $- \frac{\sqrt{3} \sqrt{2}}{2}$ into the numerator.

$\cos (θ) = \frac{- \sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.2

Cancel the common factor.

$\cos (θ) = \frac{- \sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.3

Rewrite the expression.

$\cos (θ) = - \sqrt{3} \sqrt{2} \frac{1}{\sqrt{7}}$

Step 4.5

Combine using the product rule for radicals.

$\cos (θ) = - \sqrt{2 \cdot 3} \frac{1}{\sqrt{7}}$

Step 4.6

Multiply $2$ by $3$ .

$\cos (θ) = - \sqrt{6} \frac{1}{\sqrt{7}}$

Step 4.7

Combine $\frac{1}{\sqrt{7}}$ and $\sqrt{6}$ .

$\cos (θ) = - \frac{\sqrt{6}}{\sqrt{7}}$

Step 4.8

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cos (θ) = - (\frac{\sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 4.9

Combine and simplify the denominator.

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Step 4.9.1

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.9.2

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

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.9.3

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

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.9.4

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

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.9.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.9.6

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

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Step 4.9.6.1

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

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.9.6.2

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

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.9.6.3

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

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.9.6.4

Cancel the common factor of $2$ .

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Step 4.9.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.9.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{7}$

Step 4.9.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{6} \sqrt{7}}{7}$

Step 4.10

Simplify the numerator.

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Step 4.10.1

Combine using the product rule for radicals.

$\cos (θ) = - \frac{\sqrt{6 \cdot 7}}{7}$

Step 4.10.2

Multiply $6$ by $7$ .

$\cos (θ) = - \frac{\sqrt{42}}{7}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{\sqrt{42}}{7} \approx - 0.92582009$