Trigonometry Examples

$(- 5, - 3)$

Step 1

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

Opposite : $- 3$

Adjacent : $- 5$

Step 2

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

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Step 2.1

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

$\sqrt{25 + {(- 3)}^{2}}$

Step 2.2

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

$\sqrt{25 + 9}$

Step 2.3

Add $25$ and $9$ .

$\sqrt{34}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 5}{\sqrt{34}}$ .

$\frac{- 5}{\sqrt{34}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{5}{\sqrt{34}}$

Step 4.2

Multiply $\frac{5}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\cos (θ) = - (\frac{5}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{5}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\cos (θ) = - \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.3.2

Raise $\sqrt{34}$ to the power of $1$ .

$\cos (θ) = - \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.3.3

Raise $\sqrt{34}$ to the power of $1$ .

$\cos (θ) = - \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.3.4

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

$\cos (θ) = - \frac{5 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{5 \sqrt{34}}{{\sqrt{34}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{5 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = - \frac{5 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = - \frac{5 \sqrt{34}}{34^{\frac{2}{2}}}$

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.

$\cos (θ) = - \frac{5 \sqrt{34}}{34^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{5 \sqrt{34}}{34}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{5 \sqrt{34}}{34}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{5 \sqrt{34}}{34} \approx - 0.85749292$