Trigonometry Examples

$(- 3, \sqrt{7})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 3, \sqrt{7})$ , draw the triangle between the three points $(0, 0)$ , $(- 3, 0)$ , and $(- 3, \sqrt{7})$ .

Opposite : $\sqrt{7}$

Adjacent : $- 3$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise $- 3$ to the power of $2$ .

$\sqrt{9 + {(\sqrt{7})}^{2}}$

Step 2.2

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

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{9 + {(7^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{9 + 7^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

$\sqrt{9 + 7^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{9 + 7}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Add $9$ and $7$ .

$\sqrt{16}$

Step 2.3.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 2.3.3

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

$4$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\sqrt{7}}{4}$ .

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

Step 4

Approximate the result.

$\sin (θ) = \frac{\sqrt{7}}{4} \approx 0.66143782$