Trigonometry Examples

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

Step 1

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

Opposite : $\frac{24}{25}$

Adjacent : $\frac{7}{25}$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

$\sqrt{\frac{49}{25^{2}} + {(\frac{24}{25})}^{2}}$

Step 2.3

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

$\sqrt{\frac{49}{625} + {(\frac{24}{25})}^{2}}$

Step 2.4

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

$\sqrt{\frac{49}{625} + \frac{24^{2}}{25^{2}}}$

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Add $49$ and $576$ .

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

Step 2.9

Divide $625$ by $625$ .

$\sqrt{1}$

Step 2.10

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{24}{25}}{1}$ .

$\frac{\frac{24}{25}}{1}$

Step 4

Divide $\frac{24}{25}$ by $1$ .

$\sin (θ) = \frac{24}{25}$

Step 5

Approximate the result.

$\sin (θ) = \frac{24}{25} \approx 0.96$