Trigonometry Examples

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{1}{2}, 0)$ , and $(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Opposite : $- \frac{\sqrt{3}}{2}$

Adjacent : $\frac{1}{2}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.2

One to any power is one.

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

Step 2.3

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

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

Step 2.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.5

Simplify the expression.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 2.7

Simplify the expression.

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Step 2.7.1

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

$\sqrt{\frac{1}{4} + \frac{3}{4}}$

Step 2.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 3}{4}}$

Step 2.7.3

Add $1$ and $3$ .

$\sqrt{\frac{4}{4}}$

Step 2.7.4

Divide $4$ by $4$ .

$\sqrt{1}$

Step 2.7.5

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{1}{2}}{1}$ .

$\frac{\frac{1}{2}}{1}$

Step 4

Divide $\frac{1}{2}$ by $1$ .

$\cos (θ) = \frac{1}{2}$

Step 5

Approximate the result.

$\cos (θ) = \frac{1}{2} \approx 0.5$