Precalculus Examples

$(- 8, - 15)$

Step 1

Use the conversion formulas to convert from polar coordinates to rectangular coordinates.

$x = r c o s θ$

$y = r s i n θ$

Step 2

Substitute in the known values of $r = - 8$ and $θ = - 15$ into the formulas.

$x = (- 8) \cos (- 15)$

$y = (- 8) \sin (- 15)$

Step 3

The exact value of $\cos (- 15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Tap for more steps...

Step 3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$x = - 8 \cos (15)$

$y = (- 8) \sin (- 15)$

Step 3.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$x = - 8 \cos (45 - 30)$

$y = (- 8) \sin (- 15)$

Step 3.3

Separate negation.

$x = - 8 \cos (45 - (30))$

$y = (- 8) \sin (- 15)$

Step 3.4

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$x = - 8 (\cos (45) \cos (30) + \sin (45) \sin (30))$

$y = (- 8) \sin (- 15)$

Step 3.5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = - 8 (\frac{\sqrt{2}}{2} \cdot \cos (30) + \sin (45) \sin (30))$

$y = (- 8) \sin (- 15)$

Step 3.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$x = - 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30))$

$y = (- 8) \sin (- 15)$

Step 3.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = - 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \sin (30))$

$y = (- 8) \sin (- 15)$

Step 3.8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$x = - 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (- 8) \sin (- 15)$

Step 3.9

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 3.9.1

Simplify each term.

Tap for more steps...

Step 3.9.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 3.9.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$x = - 8 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (- 8) \sin (- 15)$

Step 3.9.1.1.2

Combine using the product rule for radicals.

$x = - 8 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (- 8) \sin (- 15)$

Step 3.9.1.1.3

Multiply $2$ by $3$ .

$x = - 8 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (- 8) \sin (- 15)$

Step 3.9.1.1.4

Multiply $2$ by $2$ .

$x = - 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (- 8) \sin (- 15)$

$x = - 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (- 8) \sin (- 15)$

Step 3.9.1.2

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 3.9.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$x = - 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2})$

$y = (- 8) \sin (- 15)$

Step 3.9.1.2.2

Multiply $2$ by $2$ .

$x = - 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

$y = (- 8) \sin (- 15)$

$x = - 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

$y = (- 8) \sin (- 15)$

$x = - 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

$y = (- 8) \sin (- 15)$

Step 3.9.2

Combine the numerators over the common denominator.

$x = - 8 \frac{\sqrt{6} + \sqrt{2}}{4}$

$y = (- 8) \sin (- 15)$

$x = - 8 \frac{\sqrt{6} + \sqrt{2}}{4}$

$y = (- 8) \sin (- 15)$

$x = - 8 \frac{\sqrt{6} + \sqrt{2}}{4}$

$y = (- 8) \sin (- 15)$

Step 4

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.1

Factor $4$ out of $- 8$ .

$x = 4 (- 2) (\frac{\sqrt{6} + \sqrt{2}}{4})$

$y = (- 8) \sin (- 15)$

Step 4.2

Cancel the common factor.

$x = 4 \cdot (- 2 \frac{\sqrt{6} + \sqrt{2}}{4})$

$y = (- 8) \sin (- 15)$

Step 4.3

Rewrite the expression.

$x = - 2 (\sqrt{6} + \sqrt{2})$

$y = (- 8) \sin (- 15)$

$x = - 2 (\sqrt{6} + \sqrt{2})$

$y = (- 8) \sin (- 15)$

Step 5

Apply the distributive property.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = (- 8) \sin (- 15)$

Step 6

The exact value of $\sin (- 15)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 6.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- \sin (15))$

Step 6.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- \sin (45 - 30))$

Step 6.3

Separate negation.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- \sin (45 - (30)))$

Step 6.4

Apply the difference of angles identity.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\sin (45) \cos (30) - \cos (45) \sin (30)))$

Step 6.5

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{2}}{2} \cdot \cos (30) - \cos (45) \sin (30)))$

Step 6.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)))$

Step 6.7

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \sin (30)))$

Step 6.8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 6.9

Simplify $- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$ .

Tap for more steps...

Step 6.9.1

Simplify each term.

Tap for more steps...

Step 6.9.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 6.9.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 6.9.1.1.2

Combine using the product rule for radicals.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 6.9.1.1.3

Multiply $2$ by $3$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 6.9.1.1.4

Multiply $2$ by $2$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 6.9.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 6.9.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}))$

Step 6.9.1.2.2

Multiply $2$ by $2$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

Step 6.9.2

Combine the numerators over the common denominator.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- \frac{\sqrt{6} - \sqrt{2}}{4})$

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- \frac{\sqrt{6} - \sqrt{2}}{4})$

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 (- \frac{\sqrt{6} - \sqrt{2}}{4})$

Step 7

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.1

Move the leading negative in $- \frac{\sqrt{6} - \sqrt{2}}{4}$ into the numerator.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 8 \frac{- (\sqrt{6} - \sqrt{2})}{4}$

Step 7.2

Factor $4$ out of $- 8$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = 4 (- 2) (\frac{- (\sqrt{6} - \sqrt{2})}{4})$

Step 7.3

Cancel the common factor.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = 4 \cdot (- 2 \frac{- (\sqrt{6} - \sqrt{2})}{4})$

Step 7.4

Rewrite the expression.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 2 (- (\sqrt{6} - \sqrt{2}))$

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = - 2 (- (\sqrt{6} - \sqrt{2}))$

Step 8

Multiply $- 1$ by $- 2$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = 2 (\sqrt{6} - \sqrt{2})$

Step 9

Apply the distributive property.

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = 2 \sqrt{6} + 2 (- \sqrt{2})$

Step 10

Multiply $- 1$ by $2$ .

$x = - 2 \sqrt{6} - 2 \sqrt{2}$

$y = 2 \sqrt{6} - 2 \sqrt{2}$

Step 11

The rectangular representation of the polar point $(- 8, - 15)$ is $(- 2 \sqrt{6} - 2 \sqrt{2}, 2 \sqrt{6} - 2 \sqrt{2})$ .

$(- 2 \sqrt{6} - 2 \sqrt{2}, 2 \sqrt{6} - 2 \sqrt{2})$