Trigonometry Examples

$(\frac{3}{4}, \frac{7}{16})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{3}{4}, \frac{7}{16})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{3}{4}, 0)$ , and $(\frac{3}{4}, \frac{7}{16})$ .

Opposite : $\frac{7}{16}$

Adjacent : $\frac{3}{4}$

Step 2

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

Tap for more steps...

Step 2.1

Apply the product rule to $\frac{3}{4}$ .

$\sqrt{\frac{3^{2}}{4^{2}} + {(\frac{7}{16})}^{2}}$

Step 2.2

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

$\sqrt{\frac{9}{4^{2}} + {(\frac{7}{16})}^{2}}$

Step 2.3

Raise $4$ to the power of $2$ .

$\sqrt{\frac{9}{16} + {(\frac{7}{16})}^{2}}$

Step 2.4

Apply the product rule to $\frac{7}{16}$ .

$\sqrt{\frac{9}{16} + \frac{7^{2}}{16^{2}}}$

Step 2.5

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

$\sqrt{\frac{9}{16} + \frac{49}{16^{2}}}$

Step 2.6

Raise $16$ to the power of $2$ .

$\sqrt{\frac{9}{16} + \frac{49}{256}}$

Step 2.7

To write $\frac{9}{16}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{\frac{9}{16} \cdot \frac{16}{16} + \frac{49}{256}}$

Step 2.8

Write each expression with a common denominator of $256$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.8.1

Multiply $\frac{9}{16}$ by $\frac{16}{16}$ .

$\sqrt{\frac{9 \cdot 16}{16 \cdot 16} + \frac{49}{256}}$

Step 2.8.2

Multiply $16$ by $16$ .

$\sqrt{\frac{9 \cdot 16}{256} + \frac{49}{256}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{9 \cdot 16 + 49}{256}}$

Step 2.10

Simplify the numerator.

Tap for more steps...

Step 2.10.1

Multiply $9$ by $16$ .

$\sqrt{\frac{144 + 49}{256}}$

Step 2.10.2

Add $144$ and $49$ .

$\sqrt{\frac{193}{256}}$

Step 2.11

Rewrite $\sqrt{\frac{193}{256}}$ as $\frac{\sqrt{193}}{\sqrt{256}}$ .

$\frac{\sqrt{193}}{\sqrt{256}}$

Step 2.12

Simplify the denominator.

Tap for more steps...

Step 2.12.1

Rewrite $256$ as $16^{2}$ .

$\frac{\sqrt{193}}{\sqrt{16^{2}}}$

Step 2.12.2

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

$\frac{\sqrt{193}}{16}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{7}{16}}{\frac{\sqrt{193}}{16}}$ .

$\frac{\frac{7}{16}}{\frac{\sqrt{193}}{16}}$

Step 4

Simplify $\sin (θ)$ .

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \frac{7}{16} \cdot \frac{16}{\sqrt{193}}$

Step 4.2

Cancel the common factor of $16$ .

Tap for more steps...

Step 4.2.1

Cancel the common factor.

$\sin (θ) = \frac{7}{16} \cdot \frac{16}{\sqrt{193}}$

Step 4.2.2

Rewrite the expression.

$\sin (θ) = 7 (\frac{1}{\sqrt{193}})$

Step 4.3

Combine $7$ and $\frac{1}{\sqrt{193}}$ .

$\sin (θ) = \frac{7}{\sqrt{193}}$

Step 4.4

Multiply $\frac{7}{\sqrt{193}}$ by $\frac{\sqrt{193}}{\sqrt{193}}$ .

$\sin (θ) = \frac{7}{\sqrt{193}} \cdot \frac{\sqrt{193}}{\sqrt{193}}$

Step 4.5

Combine and simplify the denominator.

Tap for more steps...

Step 4.5.1

Multiply $\frac{7}{\sqrt{193}}$ by $\frac{\sqrt{193}}{\sqrt{193}}$ .

$\sin (θ) = \frac{7 \sqrt{193}}{\sqrt{193} \sqrt{193}}$

Step 4.5.2

Raise $\sqrt{193}$ to the power of $1$ .

$\sin (θ) = \frac{7 \sqrt{193}}{\sqrt{193} \sqrt{193}}$

Step 4.5.3

Raise $\sqrt{193}$ to the power of $1$ .

$\sin (θ) = \frac{7 \sqrt{193}}{\sqrt{193} \sqrt{193}}$

Step 4.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = \frac{7 \sqrt{193}}{{\sqrt{193}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\sin (θ) = \frac{7 \sqrt{193}}{{\sqrt{193}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{193}}^{2}$ as $193$ .

Tap for more steps...

Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{193}$ as $193^{\frac{1}{2}}$ .

$\sin (θ) = \frac{7 \sqrt{193}}{{(193^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\sin (θ) = \frac{7 \sqrt{193}}{193^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\sin (θ) = \frac{7 \sqrt{193}}{193^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.5.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{7 \sqrt{193}}{193^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{7 \sqrt{193}}{193}$

Step 4.5.6.5

Evaluate the exponent.

$\sin (θ) = \frac{7 \sqrt{193}}{193}$

Step 5

Approximate the result.

$\sin (θ) = \frac{7 \sqrt{193}}{193} \approx 0.50387102$