Trigonometry Examples

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

Step 1

To find the

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

$(0, 0)$ and

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

$(0, 0)$ ,

$(- \sqrt{7}, 0)$ , and

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

Opposite :

$3$

Adjacent :

$- \sqrt{7}$

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

Apply the product rule to

$- \sqrt{7}$ .

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

Step 2.1.2

Raise

$- 1$ to the power of

$2$ .

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

Step 2.1.3

Multiply

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

$1$ .

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

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

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

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

Step 2.2.2

Apply the power rule and multiply exponents,

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

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

Step 2.2.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 2.2.4

Cancel the common factor of

$2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Evaluate the exponent.

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise

$3$ to the power of

$2$ .

$\sqrt{7 + 9}$

Step 2.3.2

Add

$7$ and

$9$ .

$\sqrt{16}$

Step 2.3.3

Rewrite

$16$ as

$4^{2}$ .

$\sqrt{4^{2}}$

Step 2.3.4

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

$4$

Step 3

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

$\sin (θ) = \frac{3}{4}$ .

$\frac{3}{4}$

Step 4

Approximate the result.

$\sin (θ) = \frac{3}{4} \approx 0.75$