Trigonometry Examples

$(\frac{- 1}{\sqrt{10}}, \frac{- 3}{\sqrt{10}})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{- 1}{\sqrt{10}}, \frac{- 3}{\sqrt{10}})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{- 1}{\sqrt{10}}, 0)$ , and $(\frac{- 1}{\sqrt{10}}, \frac{- 3}{\sqrt{10}})$ .

Opposite : $\frac{- 3}{\sqrt{10}}$

Adjacent : $\frac{- 1}{\sqrt{10}}$

Step 2

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

Tap for more steps...

Step 2.1

Move the negative in front of the fraction.

$\sqrt{{(- \frac{1}{\sqrt{10}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.2

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sqrt{{(- (\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}))}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.3.1

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sqrt{{(- \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt{{(- \frac{\sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt{{(- \frac{\sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.4

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

$\sqrt{{(- \frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.5

Add $1$ and $1$ .

$\sqrt{{(- \frac{\sqrt{10}}{{\sqrt{10}}^{2}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{{(- \frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.6.2

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

$\sqrt{{(- \frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.6.3

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

$\sqrt{{(- \frac{\sqrt{10}}{10^{\frac{2}{2}}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.6.4.1

Cancel the common factor.

$\sqrt{{(- \frac{\sqrt{10}}{10^{\frac{2}{2}}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.6.4.2

Rewrite the expression.

$\sqrt{{(- \frac{\sqrt{10}}{10^{1}})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.3.6.5

Evaluate the exponent.

$\sqrt{{(- \frac{\sqrt{10}}{10})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.4

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

Tap for more steps...

Step 2.4.1

Apply the product rule to $- \frac{\sqrt{10}}{10}$ .

$\sqrt{{(- 1)}^{2} {(\frac{\sqrt{10}}{10})}^{2} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{10}}{10}$ .

$\sqrt{{(- 1)}^{2} \frac{{\sqrt{10}}^{2}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.5

Simplify the expression.

Tap for more steps...

Step 2.5.1

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

$\sqrt{1 \frac{{\sqrt{10}}^{2}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.5.2

Multiply $\frac{{\sqrt{10}}^{2}}{10^{2}}$ by $1$ .

$\sqrt{\frac{{\sqrt{10}}^{2}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{\frac{{(10^{\frac{1}{2}})}^{2}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.6.2

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

$\sqrt{\frac{10^{\frac{1}{2} \cdot 2}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.6.3

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

$\sqrt{\frac{10^{\frac{2}{2}}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{10^{\frac{2}{2}}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{10^{1}}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{10}{10^{2}} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.7

Raise $10$ to the power of $2$ .

$\sqrt{\frac{10}{100} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.8

Cancel the common factor of $10$ and $100$ .

Tap for more steps...

Step 2.8.1

Factor $10$ out of $10$ .

$\sqrt{\frac{10 (1)}{100} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.8.2

Cancel the common factors.

Tap for more steps...

Step 2.8.2.1

Factor $10$ out of $100$ .

$\sqrt{\frac{10 \cdot 1}{10 \cdot 10} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.8.2.2

Cancel the common factor.

$\sqrt{\frac{10 \cdot 1}{10 \cdot 10} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.8.2.3

Rewrite the expression.

$\sqrt{\frac{1}{10} + {(\frac{- 3}{\sqrt{10}})}^{2}}$

Step 2.9

Move the negative in front of the fraction.

$\sqrt{\frac{1}{10} + {(- \frac{3}{\sqrt{10}})}^{2}}$

Step 2.10

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sqrt{\frac{1}{10} + {(- (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}))}^{2}}$

Step 2.11

Combine and simplify the denominator.

Tap for more steps...

Step 2.11.1

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}})}^{2}}$

Step 2.11.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}})}^{2}}$

Step 2.11.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}})}^{2}}$

Step 2.11.4

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

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}})}^{2}}$

Step 2.11.5

Add $1$ and $1$ .

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{{\sqrt{10}}^{2}})}^{2}}$

Step 2.11.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 2.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}})}^{2}}$

Step 2.11.6.2

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

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}})}^{2}}$

Step 2.11.6.3

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

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{10^{\frac{2}{2}}})}^{2}}$

Step 2.11.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.11.6.4.1

Cancel the common factor.

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{10^{\frac{2}{2}}})}^{2}}$

Step 2.11.6.4.2

Rewrite the expression.

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{10^{1}})}^{2}}$

Step 2.11.6.5

Evaluate the exponent.

$\sqrt{\frac{1}{10} + {(- \frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.12

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

Tap for more steps...

Step 2.12.1

Apply the product rule to $- \frac{3 \sqrt{10}}{10}$ .

$\sqrt{\frac{1}{10} + {(- 1)}^{2} {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.12.2

Apply the product rule to $\frac{3 \sqrt{10}}{10}$ .

$\sqrt{\frac{1}{10} + {(- 1)}^{2} \frac{{(3 \sqrt{10})}^{2}}{10^{2}}}$

Step 2.12.3

Apply the product rule to $3 \sqrt{10}$ .

$\sqrt{\frac{1}{10} + {(- 1)}^{2} \frac{3^{2} {\sqrt{10}}^{2}}{10^{2}}}$

Step 2.13

Simplify the expression.

Tap for more steps...

Step 2.13.1

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

$\sqrt{\frac{1}{10} + 1 \frac{3^{2} {\sqrt{10}}^{2}}{10^{2}}}$

Step 2.13.2

Multiply $\frac{3^{2} {\sqrt{10}}^{2}}{10^{2}}$ by $1$ .

$\sqrt{\frac{1}{10} + \frac{3^{2} {\sqrt{10}}^{2}}{10^{2}}}$

Step 2.14

Simplify the numerator.

Tap for more steps...

Step 2.14.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{1}{10} + \frac{9 {\sqrt{10}}^{2}}{10^{2}}}$

Step 2.14.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 2.14.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{10} + \frac{9 {(10^{\frac{1}{2}})}^{2}}{10^{2}}}$

Step 2.14.2.2

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

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{\frac{1}{2} \cdot 2}}{10^{2}}}$

Step 2.14.2.3

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

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{\frac{2}{2}}}{10^{2}}}$

Step 2.14.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.14.2.4.1

Cancel the common factor.

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{\frac{2}{2}}}{10^{2}}}$

Step 2.14.2.4.2

Rewrite the expression.

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{1}}{10^{2}}}$

Step 2.14.2.5

Evaluate the exponent.

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10}{10^{2}}}$

Step 2.15

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.15.1

Raise $10$ to the power of $2$ .

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10}{100}}$

Step 2.15.2

Multiply $9$ by $10$ .

$\sqrt{\frac{1}{10} + \frac{90}{100}}$

Step 2.15.3

Cancel the common factor of $90$ and $100$ .

Tap for more steps...

Step 2.15.3.1

Factor $10$ out of $90$ .

$\sqrt{\frac{1}{10} + \frac{10 (9)}{100}}$

Step 2.15.3.2

Cancel the common factors.

Tap for more steps...

Step 2.15.3.2.1

Factor $10$ out of $100$ .

$\sqrt{\frac{1}{10} + \frac{10 \cdot 9}{10 \cdot 10}}$

Step 2.15.3.2.2

Cancel the common factor.

$\sqrt{\frac{1}{10} + \frac{10 \cdot 9}{10 \cdot 10}}$

Step 2.15.3.2.3

Rewrite the expression.

$\sqrt{\frac{1}{10} + \frac{9}{10}}$

Step 2.15.4

Simplify the expression.

Tap for more steps...

Step 2.15.4.1

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 9}{10}}$

Step 2.15.4.2

Add $1$ and $9$ .

$\sqrt{\frac{10}{10}}$

Step 2.15.4.3

Divide $10$ by $10$ .

$\sqrt{1}$

Step 2.15.4.4

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{- 1}{\sqrt{10}}}{1}$ .

$\frac{\frac{- 1}{\sqrt{10}}}{1}$

Step 4

Simplify $\cos (θ)$ .

Tap for more steps...

Step 4.1

Divide $\frac{- 1}{\sqrt{10}}$ by $1$ .

$\cos (θ) = \frac{- 1}{\sqrt{10}}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{10}}$

Step 4.3

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = - (\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 4.4

Combine and simplify the denominator.

Tap for more steps...

Step 4.4.1

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.4.2

Raise $\sqrt{10}$ to the power of $1$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.4.3

Raise $\sqrt{10}$ to the power of $1$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.4.4

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

$\cos (θ) = - \frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{\sqrt{10}}{{\sqrt{10}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{10}}{10}$

Step 4.4.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{10}}{10}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{\sqrt{10}}{10} \approx - 0.31622776$