Trigonometry Examples

$(- \sqrt{19}, 9)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \sqrt{19}, 9)$ , draw the triangle between the three points $(0, 0)$ , $(- \sqrt{19}, 0)$ , and $(- \sqrt{19}, 9)$ .

Opposite : $9$

Adjacent : $- \sqrt{19}$

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

Apply the product rule to $- \sqrt{19}$ .

$\sqrt{{(- 1)}^{2} {\sqrt{19}}^{2} + {(9)}^{2}}$

Step 2.1.2

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

$\sqrt{1 {\sqrt{19}}^{2} + {(9)}^{2}}$

Step 2.1.3

Multiply ${\sqrt{19}}^{2}$ by $1$ .

$\sqrt{{\sqrt{19}}^{2} + {(9)}^{2}}$

Step 2.2

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\sqrt{{(19^{\frac{1}{2}})}^{2} + {(9)}^{2}}$

Step 2.2.2

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

$\sqrt{19^{\frac{1}{2} \cdot 2} + {(9)}^{2}}$

Step 2.2.3

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

$\sqrt{19^{\frac{2}{2}} + {(9)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{19^{\frac{2}{2}} + {(9)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{19^{1} + {(9)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{19 + {(9)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise $9$ to the power of $2$ .

$\sqrt{19 + 81}$

Step 2.3.2

Add $19$ and $81$ .

$\sqrt{100}$

Step 2.3.3

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}}$

Step 2.3.4

Pull terms out from under the radical, assuming positive real numbers.

$10$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{9}{10}$ .

$\frac{9}{10}$

Step 4

Approximate the result.

$\sin (θ) = \frac{9}{10} \approx 0.9$