Trigonometry Examples

(\frac{\sqrt{7}}{4}, \frac{3}{4})

$(\frac{\sqrt{7}}{4}, \frac{3}{4})$

Step 1

To find the

sin (θ)

$\sin (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(\frac{\sqrt{7}}{4}, \frac{3}{4})

$(\frac{\sqrt{7}}{4}, \frac{3}{4})$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(\frac{\sqrt{7}}{4}, 0)

$(\frac{\sqrt{7}}{4}, 0)$ , and

(\frac{\sqrt{7}}{4}, \frac{3}{4})

$(\frac{\sqrt{7}}{4}, \frac{3}{4})$ .

Opposite :

\frac{3}{4}

$\frac{3}{4}$

Adjacent :

\frac{\sqrt{7}}{4}

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

Step 2

Find the hypotenuse using Pythagorean theorem

c = \sqrt{a^{2} + b^{2}}

$c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Apply the product rule to

\frac{\sqrt{7}}{4}

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

\sqrt{\frac{{\sqrt{7}}^{2}}{4^{2}} + {(\frac{3}{4})}^{2}}

$\sqrt{\frac{{\sqrt{7}}^{2}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.2

Rewrite

{\sqrt{7}}^{2}

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

7

$7$ .

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Step 2.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{7}

$\sqrt{7}$ as

7^{\frac{1}{2}}

$7^{\frac{1}{2}}$ .

\sqrt{\frac{{(7^{\frac{1}{2}})}^{2}}{4^{2}} + {(\frac{3}{4})}^{2}}

$\sqrt{\frac{{(7^{\frac{1}{2}})}^{2}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{\frac{7^{\frac{1}{2} \cdot 2}}{4^{2}} + {(\frac{3}{4})}^{2}}

$\sqrt{\frac{7^{\frac{1}{2} \cdot 2}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{7^{\frac{2}{2}}}{4^{2}} + {(\frac{3}{4})}^{2}}

$\sqrt{\frac{7^{\frac{2}{2}}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.2.4

Cancel the common factor of

2

$2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{\frac{7^{\frac{2}{2}}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{\frac{7^{1}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{\frac{7}{4^{2}} + {(\frac{3}{4})}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise

$4$ to the power of

$2$ .

$\sqrt{\frac{7}{16} + {(\frac{3}{4})}^{2}}$

Step 2.3.2

Apply the product rule to

$\frac{3}{4}$ .

$\sqrt{\frac{7}{16} + \frac{3^{2}}{4^{2}}}$

Step 2.3.3

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{7}{16} + \frac{9}{4^{2}}}$

Step 2.3.4

Raise

$4$ to the power of

$2$ .

$\sqrt{\frac{7}{16} + \frac{9}{16}}$

Step 2.3.5

Combine the numerators over the common denominator.

$\sqrt{\frac{7 + 9}{16}}$

Step 2.3.6

Add

$7$ and

$9$ .

$\sqrt{\frac{16}{16}}$

Step 2.3.7

Divide

$16$ by

$16$ .

$\sqrt{1}$

Step 2.3.8

Any root of

$1$ is

$1$ .

$1$

Step 3

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

$\sin (θ) = \frac{\frac{3}{4}}{1}$ .

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

Step 4

Divide

$\frac{3}{4}$ by

$1$ .

$\sin (θ) = \frac{3}{4}$

Step 5

Approximate the result.

$\sin (θ) = \frac{3}{4} \approx 0.75$