Trigonometry Examples

$(- 5, - \frac{3 π}{4})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 5, - \frac{3 π}{4})$ , draw the triangle between the three points $(0, 0)$ , $(- 5, 0)$ , and $(- 5, - \frac{3 π}{4})$ .

Opposite : $- \frac{3 π}{4}$

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 $2$ .

$\sqrt{25 + {(- \frac{3 π}{4})}^{2}}$

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 $- \frac{3 π}{4}$ .

$\sqrt{25 + {(- 1)}^{2} {(\frac{3 π}{4})}^{2}}$

Step 2.2.2

Apply the product rule to $\frac{3 π}{4}$ .

$\sqrt{25 + {(- 1)}^{2} \frac{{(3 π)}^{2}}{4^{2}}}$

Step 2.2.3

Apply the product rule to $3 π$ .

$\sqrt{25 + {(- 1)}^{2} \frac{3^{2} π^{2}}{4^{2}}}$

Step 2.3

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

$\sqrt{25 + 1 \frac{3^{2} π^{2}}{4^{2}}}$

Step 2.4

Multiply $\frac{3^{2} π^{2}}{4^{2}}$ by $1$ .

$\sqrt{25 + \frac{3^{2} π^{2}}{4^{2}}}$

Step 2.5

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

$\sqrt{25 + \frac{9 π^{2}}{4^{2}}}$

Step 2.6

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

$\sqrt{25 + \frac{9 π^{2}}{16}}$

Step 2.7

To write $25$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{25 \cdot \frac{16}{16} + \frac{9 π^{2}}{16}}$

Step 2.8

Combine $25$ and $\frac{16}{16}$ .

$\sqrt{\frac{25 \cdot 16}{16} + \frac{9 π^{2}}{16}}$

Step 2.9

Simplify the expression.

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Step 2.9.1

Combine the numerators over the common denominator.

$\sqrt{\frac{25 \cdot 16 + 9 π^{2}}{16}}$

Step 2.9.2

Multiply $25$ by $16$ .

$\sqrt{\frac{400 + 9 π^{2}}{16}}$

Step 2.10

Rewrite $\sqrt{\frac{400 + 9 π^{2}}{16}}$ as $\frac{\sqrt{400 + 9 π^{2}}}{\sqrt{16}}$ .

$\frac{\sqrt{400 + 9 π^{2}}}{\sqrt{16}}$

Step 2.11

Simplify the denominator.

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Step 2.11.1

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{400 + 9 π^{2}}}{\sqrt{4^{2}}}$

Step 2.11.2

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

$\frac{\sqrt{400 + 9 π^{2}}}{4}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 5}{\frac{\sqrt{400 + 9 π^{2}}}{4}}$ .

$\frac{- 5}{\frac{\sqrt{400 + 9 π^{2}}}{4}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - 5 \frac{4}{\sqrt{400 + 9 π^{2}}}$

Step 4.2

Multiply $\frac{4}{\sqrt{400 + 9 π^{2}}}$ by $\frac{\sqrt{400 + 9 π^{2}}}{\sqrt{400 + 9 π^{2}}}$ .

$\cos (θ) = - 5 (\frac{4}{\sqrt{400 + 9 π^{2}}} \cdot \frac{\sqrt{400 + 9 π^{2}}}{\sqrt{400 + 9 π^{2}}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{4}{\sqrt{400 + 9 π^{2}}}$ by $\frac{\sqrt{400 + 9 π^{2}}}{\sqrt{400 + 9 π^{2}}}$ .

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{\sqrt{400 + 9 π^{2}} \sqrt{400 + 9 π^{2}}}$

Step 4.3.2

Raise $\sqrt{400 + 9 π^{2}}$ to the power of $1$ .

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{\sqrt{400 + 9 π^{2}} \sqrt{400 + 9 π^{2}}}$

Step 4.3.3

Raise $\sqrt{400 + 9 π^{2}}$ to the power of $1$ .

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{\sqrt{400 + 9 π^{2}} \sqrt{400 + 9 π^{2}}}$

Step 4.3.4

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

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{{\sqrt{400 + 9 π^{2}}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{{\sqrt{400 + 9 π^{2}}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{400 + 9 π^{2}}}^{2}$ as $400 + 9 π^{2}$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{400 + 9 π^{2}}$ as ${(400 + 9 π^{2})}^{\frac{1}{2}}$ .

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{{({(400 + 9 π^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{{(400 + 9 π^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{{(400 + 9 π^{2})}^{\frac{2}{2}}}$

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.

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{{(400 + 9 π^{2})}^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}}$

Step 4.3.6.5

Simplify.

$\cos (θ) = - 5 \frac{4 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}}$

Step 4.4

Multiply $- 5 \frac{4 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}}$ .

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Step 4.4.1

Combine $- 5$ and $\frac{4 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}}$ .

$\cos (θ) = \frac{- 5 (4 \sqrt{400 + 9 π^{2}})}{400 + 9 π^{2}}$

Step 4.4.2

Multiply $4$ by $- 5$ .

$\cos (θ) = \frac{- 20 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}}$

Step 4.5

Move the negative in front of the fraction.

$\cos (θ) = - \frac{20 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{20 \sqrt{400 + 9 π^{2}}}{400 + 9 π^{2}} \approx - 0.90459181$