Trigonometry Examples

$(1, - 8)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(1, - 8)$ , draw the triangle between the three points $(0, 0)$ , $(1, 0)$ , and $(1, - 8)$ .

Opposite : $- 8$

Adjacent : $1$

Step 2

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

Tap for more steps...

Step 2.1

One to any power is one.

$\sqrt{1 + {(- 8)}^{2}}$

Step 2.2

Raise $- 8$ to the power of $2$ .

$\sqrt{1 + 64}$

Step 2.3

Add $1$ and $64$ .

$\sqrt{65}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 8}{\sqrt{65}}$ .

$\frac{- 8}{\sqrt{65}}$

Step 4

Simplify $\sin (θ)$ .

Tap for more steps...

Step 4.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{8}{\sqrt{65}}$

Step 4.2

Multiply $\frac{8}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\sin (θ) = - (\frac{8}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}})$

Step 4.3

Combine and simplify the denominator.

Tap for more steps...

Step 4.3.1

Multiply $\frac{8}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\sin (θ) = - \frac{8 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Step 4.3.2

Raise $\sqrt{65}$ to the power of $1$ .

$\sin (θ) = - \frac{8 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Step 4.3.3

Raise $\sqrt{65}$ to the power of $1$ .

$\sin (θ) = - \frac{8 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Step 4.3.4

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

$\sin (θ) = - \frac{8 \sqrt{65}}{{\sqrt{65}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{8 \sqrt{65}}{{\sqrt{65}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{65}}^{2}$ as $65$ .

Tap for more steps...

Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{8 \sqrt{65}}{{(65^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\sin (θ) = - \frac{8 \sqrt{65}}{65^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\sin (θ) = - \frac{8 \sqrt{65}}{65^{\frac{2}{2}}}$

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.

$\sin (θ) = - \frac{8 \sqrt{65}}{65^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{8 \sqrt{65}}{65}$

Step 4.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{8 \sqrt{65}}{65}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{8 \sqrt{65}}{65} \approx - 0.99227787$