Trigonometry Examples

$(- 7, - 8)$

Step 1

To find the

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

$(0, 0)$ and

$(- 7, - 8)$ , draw the triangle between the three points

$(0, 0)$ ,

$(- 7, 0)$ , and

$(- 7, - 8)$ .

Opposite :

$- 8$

Adjacent :

$- 7$

Step 2

Find the hypotenuse using Pythagorean theorem

$c = \sqrt{a^{2} + b^{2}}$ .

Tap for more steps...

Step 2.1

Raise

$- 7$ to the power of

$2$ .

$\sqrt{49 + {(- 8)}^{2}}$

Step 2.2

Raise

$- 8$ to the power of

$2$ .

$\sqrt{49 + 64}$

Step 2.3

Add

$49$ and

$64$ .

$\sqrt{113}$

Step 3

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

$\sin (θ) = \frac{- 8}{\sqrt{113}}$ .

$\frac{- 8}{\sqrt{113}}$

Step 4

Simplify

$\sin (θ)$ .

Tap for more steps...

Step 4.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{8}{\sqrt{113}}$

Step 4.2

Multiply

$\frac{8}{\sqrt{113}}$ by

$\frac{\sqrt{113}}{\sqrt{113}}$ .

$\sin (θ) = - (\frac{8}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}})$

Step 4.3

Combine and simplify the denominator.

Tap for more steps...

Step 4.3.1

Multiply

$\frac{8}{\sqrt{113}}$ by

$\frac{\sqrt{113}}{\sqrt{113}}$ .

$\sin (θ) = - \frac{8 \sqrt{113}}{\sqrt{113} \sqrt{113}}$

Step 4.3.2

Raise

$\sqrt{113}$ to the power of

$1$ .

$\sin (θ) = - \frac{8 \sqrt{113}}{\sqrt{113} \sqrt{113}}$

Step 4.3.3

Raise

$\sqrt{113}$ to the power of

$1$ .

$\sin (θ) = - \frac{8 \sqrt{113}}{\sqrt{113} \sqrt{113}}$

Step 4.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = - \frac{8 \sqrt{113}}{{\sqrt{113}}^{1 + 1}}$

Step 4.3.5

Add

$1$ and

$1$ .

$\sin (θ) = - \frac{8 \sqrt{113}}{{\sqrt{113}}^{2}}$

Step 4.3.6

Rewrite

${\sqrt{113}}^{2}$ as

$113$ .

Tap for more steps...

Step 4.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{113}$ as

$113^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{8 \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

Apply the power rule and multiply exponents,

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

$\sin (θ) = - \frac{8 \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sin (θ) = - \frac{8 \sqrt{113}}{113^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 4.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{8 \sqrt{113}}{113^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{8 \sqrt{113}}{113}$

Step 4.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{8 \sqrt{113}}{113}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{8 \sqrt{113}}{113} \approx - 0.75257669$