Trigonometry Examples

(3, 8)

$(3, 8)$

Step 1

To find the

sin (θ)

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

(0, 0)

$(0, 0)$ and

(3, 8)

$(3, 8)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(3, 0)

$(3, 0)$ , and

(3, 8)

$(3, 8)$ .

Opposite :

8

$8$

Adjacent :

3

$3$

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

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{9 + {(8)}^{2}}

$\sqrt{9 + {(8)}^{2}}$

Step 2.2

Raise

8

$8$ to the power of

2

$2$ .

\sqrt{9 + 64}

$\sqrt{9 + 64}$

Step 2.3

Add

9

$9$ and

64

$64$ .

\sqrt{73}

$\sqrt{73}$

\sqrt{73}

$\sqrt{73}$

Step 3

sin (θ) = \frac{Opposite}{Hypotenuse}

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

sin (θ) = \frac{8}{\sqrt{73}}

$\sin (θ) = \frac{8}{\sqrt{73}}$ .

\frac{8}{\sqrt{73}}

$\frac{8}{\sqrt{73}}$

Step 4

Simplify

sin (θ)

$\sin (θ)$ .

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Step 4.1

Multiply

\frac{8}{\sqrt{73}}

$\frac{8}{\sqrt{73}}$ by

\frac{\sqrt{73}}{\sqrt{73}}

$\frac{\sqrt{73}}{\sqrt{73}}$ .

sin (θ) = \frac{8}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}}

$\sin (θ) = \frac{8}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply

\frac{8}{\sqrt{73}}

$\frac{8}{\sqrt{73}}$ by

\frac{\sqrt{73}}{\sqrt{73}}

$\frac{\sqrt{73}}{\sqrt{73}}$ .

sin (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}

$\sin (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Step 4.2.2

Raise

\sqrt{73}

$\sqrt{73}$ to the power of

1

$1$ .

sin (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}

$\sin (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Step 4.2.3

Raise

\sqrt{73}

$\sqrt{73}$ to the power of

1

$1$ .

sin (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}

$\sin (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Step 4.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

sin (θ) = \frac{8 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}

$\sin (θ) = \frac{8 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}$

Step 4.2.5

Add

1

$1$ and

1

$1$ .

sin (θ) = \frac{8 \sqrt{73}}{{\sqrt{73}}^{2}}

$\sin (θ) = \frac{8 \sqrt{73}}{{\sqrt{73}}^{2}}$

Step 4.2.6

Rewrite

{\sqrt{73}}^{2}

${\sqrt{73}}^{2}$ as

73

$73$ .

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Step 4.2.6.1

Use

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

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

\sqrt{73}

$\sqrt{73}$ as

73^{\frac{1}{2}}

$73^{\frac{1}{2}}$ .

sin (θ) = \frac{8 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}

$\sin (θ) = \frac{8 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

Apply the power rule and multiply exponents,

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

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

sin (θ) = \frac{8 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}

$\sin (θ) = \frac{8 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

sin (θ) = \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}

$\sin (θ) = \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of

2

$2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{8 \sqrt{73}}{73}$

Step 4.2.6.5

Evaluate the exponent.

$\sin (θ) = \frac{8 \sqrt{73}}{73}$

Step 5

Approximate the result.

$\sin (θ) = \frac{8 \sqrt{73}}{73} \approx 0.93632917$

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