Trigonometry Examples

$r = 10 \sin (θ)$

Step 1

Since $\sin (θ) = \frac{y}{r}$ , replace $\sin (θ)$ with $\frac{y}{r}$ .

$r = 10 \frac{y}{r}$

Step 2

Multiply both sides by $r$ .

Tap for more steps...

Step 2.1

Multiply both sides by $r$ .

$r \cdot r = r (10 \frac{y}{r})$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Multiply $r$ by $r$ .

$r^{2} = r (10 \frac{y}{r})$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify $r (10 \frac{y}{r})$ .

Tap for more steps...

Step 2.3.1.1

Rewrite using the commutative property of multiplication.

$r^{2} = 10 r \frac{y}{r}$

Step 2.3.1.2

Cancel the common factor of $r$ .

Tap for more steps...

Step 2.3.1.2.1

Factor $r$ out of $10 r$ .

$r^{2} = r \cdot 10 \frac{y}{r}$

Step 2.3.1.2.2

Cancel the common factor.

$r^{2} = r \cdot 10 \frac{y}{r}$

Step 2.3.1.2.3

Rewrite the expression.

$r^{2} = 10 y$

Step 3

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = 10 y$

Step 4

Write in standard form.

Tap for more steps...

Step 4.1

Subtract $10 y$ from both sides of the equation.

$x^{2} + y^{2} - 10 y = 0$

Step 4.2

Complete the square for $y^{2} - 10 y$ .

Tap for more steps...

Step 4.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 10$

$c = 0$

Step 4.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 4.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 4.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 10}{2 \cdot 1}$

Step 4.2.3.2

Cancel the common factor of $- 10$ and $2$ .

Tap for more steps...

Step 4.2.3.2.1

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Step 4.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 5}{2 (1)}$

Step 4.2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Step 4.2.3.2.2.3

Rewrite the expression.

$d = \frac{- 5}{1}$

Step 4.2.3.2.2.4

Divide $- 5$ by $1$ .

$d = - 5$

Step 4.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 4.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

Step 4.2.4.2

Simplify the right side.

Tap for more steps...

Step 4.2.4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.4.2.1.1

Raise $- 10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Step 4.2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Step 4.2.4.2.1.3

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Step 4.2.4.2.1.4

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Step 4.2.4.2.2

Subtract $25$ from $0$ .

$e = - 25$

Step 4.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 5)}^{2} - 25$ .

${(y - 5)}^{2} - 25$

Step 4.3

Substitute ${(y - 5)}^{2} - 25$ for $y^{2} - 10 y$ in the equation $x^{2} + y^{2} - 10 y = 0$ .

$x^{2} + {(y - 5)}^{2} - 25 = 0$

Step 4.4

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

$x^{2} + {(y - 5)}^{2} = 0 + 25$

Step 4.5

Add $0$ and $25$ .

$x^{2} + {(y - 5)}^{2} = 25$