Precalculus Examples

$(- \frac{24}{25}, - \frac{7}{25})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{24}{25}, - \frac{7}{25})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{24}{25}, 0)$ , and $(- \frac{24}{25}, - \frac{7}{25})$ .

Opposite : $- \frac{7}{25}$

Adjacent : $- \frac{24}{25}$

Step 2

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

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $- \frac{24}{25}$ .

$\sqrt{{(- 1)}^{2} {(\frac{24}{25})}^{2} + {(- \frac{7}{25})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{24}{25}$ .

$\sqrt{{(- 1)}^{2} \frac{24^{2}}{25^{2}} + {(- \frac{7}{25})}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{24^{2}}{25^{2}} + {(- \frac{7}{25})}^{2}}$

Step 2.3

Multiply $\frac{24^{2}}{25^{2}}$ by $1$ .

$\sqrt{\frac{24^{2}}{25^{2}} + {(- \frac{7}{25})}^{2}}$

Step 2.4

Raise $24$ to the power of $2$ .

$\sqrt{\frac{576}{25^{2}} + {(- \frac{7}{25})}^{2}}$

Step 2.5

Raise $25$ to the power of $2$ .

$\sqrt{\frac{576}{625} + {(- \frac{7}{25})}^{2}}$

Step 2.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.6.1

Apply the product rule to $- \frac{7}{25}$ .

$\sqrt{\frac{576}{625} + {(- 1)}^{2} {(\frac{7}{25})}^{2}}$

Step 2.6.2

Apply the product rule to $\frac{7}{25}$ .

$\sqrt{\frac{576}{625} + {(- 1)}^{2} \frac{7^{2}}{25^{2}}}$

Step 2.7

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

$\sqrt{\frac{576}{625} + 1 \frac{7^{2}}{25^{2}}}$

Step 2.8

Multiply $\frac{7^{2}}{25^{2}}$ by $1$ .

$\sqrt{\frac{576}{625} + \frac{7^{2}}{25^{2}}}$

Step 2.9

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

$\sqrt{\frac{576}{625} + \frac{49}{25^{2}}}$

Step 2.10

Raise $25$ to the power of $2$ .

$\sqrt{\frac{576}{625} + \frac{49}{625}}$

Step 2.11

Combine the numerators over the common denominator.

$\sqrt{\frac{576 + 49}{625}}$

Step 2.12

Add $576$ and $49$ .

$\sqrt{\frac{625}{625}}$

Step 2.13

Divide $625$ by $625$ .

$\sqrt{1}$

Step 2.14

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{7}{25}}{1}$ .

$\frac{- \frac{7}{25}}{1}$

Step 4

Divide $- \frac{7}{25}$ by $1$ .

$\sin (θ) = - \frac{7}{25}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{7}{25} \approx - 0.28$