Trigonometry Examples

$(4, 4 \sqrt{3})$

Step 1

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

Opposite : $4 \sqrt{3}$

Adjacent : $4$

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 $4$ to the power of $2$ .

$\sqrt{16 + {(4 \sqrt{3})}^{2}}$

Step 2.1.2

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

$\sqrt{16 + 4^{2} {\sqrt{3}}^{2}}$

Step 2.1.3

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

$\sqrt{16 + 16 {\sqrt{3}}^{2}}$

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 $3^{\frac{1}{2}}$ .

$\sqrt{16 + 16 {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{16 + 16 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{16 + 16 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{16 + 16 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{16 + 16 \cdot 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{16 + 16 \cdot 3}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $16$ by $3$ .

$\sqrt{16 + 48}$

Step 2.3.2

Add $16$ and $48$ .

$\sqrt{64}$

Step 2.3.3

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2}}$

Step 2.3.4

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

$8$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{4}{8}$ .

$\frac{4}{8}$

Step 4

Cancel the common factor of $4$ and $8$ .

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Step 4.1

Factor $4$ out of $4$ .

$\cos (θ) = \frac{4 (1)}{8}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $8$ .

$\cos (θ) = \frac{4 \cdot 1}{4 \cdot 2}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{4 \cdot 1}{4 \cdot 2}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = \frac{1}{2}$

Step 5

Approximate the result.

$\cos (θ) = \frac{1}{2} \approx 0.5$