Trigonometry Examples

$(- 8, 8 \sqrt{3})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 8, 8 \sqrt{3})$ , draw the triangle between the three points $(0, 0)$ , $(- 8, 0)$ , and $(- 8, 8 \sqrt{3})$ .

Opposite : $8 \sqrt{3}$

Adjacent : $- 8$

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

Raise $- 8$ to the power of $2$ .

$\sqrt{64 + {(8 \sqrt{3})}^{2}}$

Step 2.1.2

Apply the product rule to $8 \sqrt{3}$ .

$\sqrt{64 + 8^{2} {\sqrt{3}}^{2}}$

Step 2.1.3

Raise $8$ to the power of $2$ .

$\sqrt{64 + 64 {\sqrt{3}}^{2}}$

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{64 + 64 {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{64 + 64 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{64 + 64 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{64 + 64 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{64 + 64 \cdot 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{64 + 64 \cdot 3}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $64$ by $3$ .

$\sqrt{64 + 192}$

Step 2.3.2

Add $64$ and $192$ .

$\sqrt{256}$

Step 2.3.3

Rewrite $256$ as $16^{2}$ .

$\sqrt{16^{2}}$

Step 2.3.4

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

$16$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 8}{16}$ .

$\frac{- 8}{16}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Cancel the common factor of $- 8$ and $16$ .

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Step 4.1.1

Factor $8$ out of $- 8$ .

$\cos (θ) = \frac{8 (- 1)}{16}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $8$ out of $16$ .

$\cos (θ) = \frac{8 \cdot - 1}{8 \cdot 2}$

Step 4.1.2.2

Cancel the common factor.

$\cos (θ) = \frac{8 \cdot - 1}{8 \cdot 2}$

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{- 1}{2}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{2}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{1}{2} \approx - 0.5$