Trigonometry Examples

$(4, - 2 \sqrt{3})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(4, - 2 \sqrt{3})$ , draw the triangle between the three points $(0, 0)$ , $(4, 0)$ , and $(4, - 2 \sqrt{3})$ .

Opposite : $- 2 \sqrt{3}$

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 $2$ .

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

Step 2.1.2

Apply the product rule to $- 2 \sqrt{3}$ .

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

Step 2.1.3

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

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

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

$\sqrt{16 + 4 {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{16 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

$\sqrt{16 + 4 \cdot 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{16 + 4 \cdot 3}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $4$ by $3$ .

$\sqrt{16 + 12}$

Step 2.3.2

Add $16$ and $12$ .

$\sqrt{28}$

Step 2.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.4.1

Factor $4$ out of $28$ .

$\sqrt{4 (7)}$

Step 2.4.2

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

$\sqrt{2^{2} \cdot 7}$

Step 2.5

Pull terms out from under the radical.

$2 \sqrt{7}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{4}{2 \sqrt{7}}$ .

$\frac{4}{2 \sqrt{7}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Cancel the common factor of $4$ and $2$ .

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Step 4.1.1

Factor $2$ out of $4$ .

$\cos (θ) = \frac{2 \cdot 2}{2 \sqrt{7}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2 \sqrt{7}$ .

$\cos (θ) = \frac{2 \cdot 2}{2 (\sqrt{7})}$

Step 4.1.2.2

Cancel the common factor.

$\cos (θ) = \frac{2 \cdot 2}{2 \sqrt{7}}$

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{2}{\sqrt{7}}$

Step 4.2

Multiply $\frac{2}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cos (θ) = \frac{2}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{2}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cos (θ) = \frac{2 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$\cos (θ) = \frac{2 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$\cos (θ) = \frac{2 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.3.4

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

$\cos (θ) = \frac{2 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = \frac{2 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\cos (θ) = \frac{2 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = \frac{2 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = \frac{2 \sqrt{7}}{7^{\frac{2}{2}}}$

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.

$\cos (θ) = \frac{2 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{2 \sqrt{7}}{7}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = \frac{2 \sqrt{7}}{7}$

Step 5

Approximate the result.

$\cos (θ) = \frac{2 \sqrt{7}}{7} \approx 0.75592894$