Trigonometry Examples

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

Step 1

To find the

$\sin (θ)$ 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

Use the power rule

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

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Step 2.1.1

Apply the product rule to

$- \frac{1}{2}$ .

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

Step 2.1.2

Apply the product rule to

$\frac{1}{2}$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Raise

$- 1$ to the power of

$2$ .

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

Step 2.2.2

Multiply

$\frac{1^{2}}{2^{2}}$ by

$1$ .

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

Step 2.2.3

One to any power is one.

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

Step 2.2.4

Raise

$2$ to the power of

$2$ .

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

Step 2.2.5

Apply the product rule to

$\frac{\sqrt{3}}{2}$ .

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

Step 2.3

Rewrite

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

$3$ .

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Step 2.3.1

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 2.3.2

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 2.3.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 2.3.4

Cancel the common factor of

$2$ .

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Step 2.3.4.1

Cancel the common factor.

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

Step 2.3.4.2

Rewrite the expression.

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

Step 2.3.5

Evaluate the exponent.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Raise

$2$ to the power of

$2$ .

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

Step 2.4.2

Combine the numerators over the common denominator.

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

Step 2.4.3

Add

$1$ and

$3$ .

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

Step 2.4.4

Divide

$4$ by

$4$ .

$\sqrt{1}$

Step 2.4.5

Any root of

$1$ is

$1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore

$\sin (θ) = \frac{\frac{\sqrt{3}}{2}}{1}$ .

$\frac{\frac{\sqrt{3}}{2}}{1}$

Step 4

Divide

$\frac{\sqrt{3}}{2}$ by

$1$ .

$\sin (θ) = \frac{\sqrt{3}}{2}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{3}}{2} \approx 0.8660254$