Trigonometry Examples

$(- 7, 6 \sqrt{5})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 7, 6 \sqrt{5})$ , draw the triangle between the three points $(0, 0)$ , $(- 7, 0)$ , and $(- 7, 6 \sqrt{5})$ .

Opposite : $6 \sqrt{5}$

Adjacent : $- 7$

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 $- 7$ to the power of $2$ .

$\sqrt{49 + {(6 \sqrt{5})}^{2}}$

Step 2.1.2

Apply the product rule to $6 \sqrt{5}$ .

$\sqrt{49 + 6^{2} {\sqrt{5}}^{2}}$

Step 2.1.3

Raise $6$ to the power of $2$ .

$\sqrt{49 + 36 {\sqrt{5}}^{2}}$

Step 2.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{49 + 36 {(5^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{49 + 36 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{49 + 36 \cdot 5^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{49 + 36 \cdot 5^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{49 + 36 \cdot 5^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{49 + 36 \cdot 5}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $36$ by $5$ .

$\sqrt{49 + 180}$

Step 2.3.2

Add $49$ and $180$ .

$\sqrt{229}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{6 \sqrt{5}}{\sqrt{229}}$ .

$\frac{6 \sqrt{5}}{\sqrt{229}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{6 \sqrt{5}}{\sqrt{229}}$ by $\frac{\sqrt{229}}{\sqrt{229}}$ .

$\sin (θ) = \frac{6 \sqrt{5}}{\sqrt{229}} \cdot \frac{\sqrt{229}}{\sqrt{229}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{6 \sqrt{5}}{\sqrt{229}}$ by $\frac{\sqrt{229}}{\sqrt{229}}$ .

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{\sqrt{229} \sqrt{229}}$

Step 4.2.2

Raise $\sqrt{229}$ to the power of $1$ .

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{\sqrt{229} \sqrt{229}}$

Step 4.2.3

Raise $\sqrt{229}$ to the power of $1$ .

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{\sqrt{229} \sqrt{229}}$

Step 4.2.4

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

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{{\sqrt{229}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{{\sqrt{229}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{229}}^{2}$ as $229$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{229}$ as $229^{\frac{1}{2}}$ .

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{{(229^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{229^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{229^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{229^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{229}$

Step 4.2.6.5

Evaluate the exponent.

$\sin (θ) = \frac{6 \sqrt{5} \sqrt{229}}{229}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Combine using the product rule for radicals.

$\sin (θ) = \frac{6 \sqrt{229 \cdot 5}}{229}$

Step 4.3.2

Multiply $229$ by $5$ .

$\sin (θ) = \frac{6 \sqrt{1145}}{229}$

Step 5

Approximate the result.

$\sin (θ) = \frac{6 \sqrt{1145}}{229} \approx 0.88658118$