Trigonometry Examples

$(\frac{1}{2}, \frac{4}{9})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{1}{2}, \frac{4}{9})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{1}{2}, 0)$ , and $(\frac{1}{2}, \frac{4}{9})$ .

Opposite : $\frac{4}{9}$

Adjacent : $\frac{1}{2}$

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 $\frac{1}{2}$ .

$\sqrt{\frac{1^{2}}{2^{2}} + {(\frac{4}{9})}^{2}}$

Step 2.2

One to any power is one.

$\sqrt{\frac{1}{2^{2}} + {(\frac{4}{9})}^{2}}$

Step 2.3

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

$\sqrt{\frac{1}{4} + {(\frac{4}{9})}^{2}}$

Step 2.4

Apply the product rule to $\frac{4}{9}$ .

$\sqrt{\frac{1}{4} + \frac{4^{2}}{9^{2}}}$

Step 2.5

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

$\sqrt{\frac{1}{4} + \frac{16}{9^{2}}}$

Step 2.6

Raise $9$ to the power of $2$ .

$\sqrt{\frac{1}{4} + \frac{16}{81}}$

Step 2.7

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\sqrt{\frac{1}{4} \cdot \frac{81}{81} + \frac{16}{81}}$

Step 2.8

To write $\frac{16}{81}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{4} \cdot \frac{81}{81} + \frac{16}{81} \cdot \frac{4}{4}}$

Step 2.9

Write each expression with a common denominator of $324$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.9.1

Multiply $\frac{1}{4}$ by $\frac{81}{81}$ .

$\sqrt{\frac{81}{4 \cdot 81} + \frac{16}{81} \cdot \frac{4}{4}}$

Step 2.9.2

Multiply $4$ by $81$ .

$\sqrt{\frac{81}{324} + \frac{16}{81} \cdot \frac{4}{4}}$

Step 2.9.3

Multiply $\frac{16}{81}$ by $\frac{4}{4}$ .

$\sqrt{\frac{81}{324} + \frac{16 \cdot 4}{81 \cdot 4}}$

Step 2.9.4

Multiply $81$ by $4$ .

$\sqrt{\frac{81}{324} + \frac{16 \cdot 4}{324}}$

Step 2.10

Combine the numerators over the common denominator.

$\sqrt{\frac{81 + 16 \cdot 4}{324}}$

Step 2.11

Simplify the numerator.

Tap for more steps...

Step 2.11.1

Multiply $16$ by $4$ .

$\sqrt{\frac{81 + 64}{324}}$

Step 2.11.2

Add $81$ and $64$ .

$\sqrt{\frac{145}{324}}$

Step 2.12

Rewrite $\sqrt{\frac{145}{324}}$ as $\frac{\sqrt{145}}{\sqrt{324}}$ .

$\frac{\sqrt{145}}{\sqrt{324}}$

Step 2.13

Simplify the denominator.

Tap for more steps...

Step 2.13.1

Rewrite $324$ as $18^{2}$ .

$\frac{\sqrt{145}}{\sqrt{18^{2}}}$

Step 2.13.2

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

$\frac{\sqrt{145}}{18}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{4}{9}}{\frac{\sqrt{145}}{18}}$ .

$\frac{\frac{4}{9}}{\frac{\sqrt{145}}{18}}$

Step 4

Simplify $\sin (θ)$ .

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \frac{4}{9} \cdot \frac{18}{\sqrt{145}}$

Step 4.2

Cancel the common factor of $9$ .

Tap for more steps...

Step 4.2.1

Factor $9$ out of $18$ .

$\sin (θ) = \frac{4}{9} \cdot \frac{9 (2)}{\sqrt{145}}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{4}{9} \cdot \frac{9 \cdot 2}{\sqrt{145}}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = 4 (\frac{2}{\sqrt{145}})$

Step 4.3

Combine $4$ and $\frac{2}{\sqrt{145}}$ .

$\sin (θ) = \frac{4 \cdot 2}{\sqrt{145}}$

Step 4.4

Multiply $4$ by $2$ .

$\sin (θ) = \frac{8}{\sqrt{145}}$

Step 4.5

Multiply $\frac{8}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\sin (θ) = \frac{8}{\sqrt{145}} \cdot \frac{\sqrt{145}}{\sqrt{145}}$

Step 4.6

Combine and simplify the denominator.

Tap for more steps...

Step 4.6.1

Multiply $\frac{8}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\sin (θ) = \frac{8 \sqrt{145}}{\sqrt{145} \sqrt{145}}$

Step 4.6.2

Raise $\sqrt{145}$ to the power of $1$ .

$\sin (θ) = \frac{8 \sqrt{145}}{\sqrt{145} \sqrt{145}}$

Step 4.6.3

Raise $\sqrt{145}$ to the power of $1$ .

$\sin (θ) = \frac{8 \sqrt{145}}{\sqrt{145} \sqrt{145}}$

Step 4.6.4

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

$\sin (θ) = \frac{8 \sqrt{145}}{{\sqrt{145}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = \frac{8 \sqrt{145}}{{\sqrt{145}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{145}}^{2}$ as $145$ .

Tap for more steps...

Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{145}$ as $145^{\frac{1}{2}}$ .

$\sin (θ) = \frac{8 \sqrt{145}}{{(145^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = \frac{8 \sqrt{145}}{145^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = \frac{8 \sqrt{145}}{145^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.6.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{8 \sqrt{145}}{145^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{8 \sqrt{145}}{145}$

Step 4.6.6.5

Evaluate the exponent.

$\sin (θ) = \frac{8 \sqrt{145}}{145}$

Step 5

Approximate the result.

$\sin (θ) = \frac{8 \sqrt{145}}{145} \approx 0.66436383$