Trigonometry Examples

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

Step 1

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

Opposite : $\frac{- \sqrt{2}}{2}$

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

Step 2

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

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Simplify the expression.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Evaluate the exponent.

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

Step 2.5

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

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

Step 2.6

Cancel the common factor of $2$ and $4$ .

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Step 2.6.1

Factor $2$ out of $2$ .

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $4$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

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

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

Step 2.9

Simplify the expression.

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Step 2.9.1

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

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

Step 2.9.2

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

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

Step 2.10

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

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Step 2.10.1

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

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

Step 2.10.2

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

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

Step 2.10.3

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

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

Step 2.10.4

Cancel the common factor of $2$ .

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Step 2.10.4.1

Cancel the common factor.

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

Step 2.10.4.2

Rewrite the expression.

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

Step 2.10.5

Evaluate the exponent.

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

Step 2.11

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

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

Step 2.12

Cancel the common factor of $2$ and $4$ .

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Step 2.12.1

Factor $2$ out of $2$ .

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

Step 2.12.2

Cancel the common factors.

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Step 2.12.2.1

Factor $2$ out of $4$ .

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

Step 2.12.2.2

Cancel the common factor.

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

Step 2.12.2.3

Rewrite the expression.

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

Step 2.13

Simplify the expression.

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Step 2.13.1

Combine the numerators over the common denominator.

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

Step 2.13.2

Add $1$ and $1$ .

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

Step 2.13.3

Divide $2$ by $2$ .

$\sqrt{1}$

Step 2.13.4

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{- \sqrt{2}}{2}}{1}$ .

$\frac{\frac{- \sqrt{2}}{2}}{1}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Divide $\frac{- \sqrt{2}}{2}$ by $1$ .

$\cos (θ) = \frac{- \sqrt{2}}{2}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{\sqrt{2}}{2} \approx - 0.70710678$