Trigonometry Examples

$(- 4, \frac{7 π}{4})$

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 $r = - 4$ and $θ = \frac{7 π}{4}$ into the formulas.

$x = (- 4) \cos (\frac{7 π}{4})$

$y = (- 4) \sin (\frac{7 π}{4})$

Step 3

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

$x = - 4 \cos (\frac{π}{4})$

$y = (- 4) \sin (\frac{7 π}{4})$

Step 4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$x = - 4 \frac{\sqrt{2}}{2}$

$y = (- 4) \sin (\frac{7 π}{4})$

Step 5

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1

Factor $2$ out of $- 4$ .

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

$y = (- 4) \sin (\frac{7 π}{4})$

Step 5.2

Cancel the common factor.

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

$y = (- 4) \sin (\frac{7 π}{4})$

Step 5.3

Rewrite the expression.

$x = - 2 \sqrt{2}$

$y = (- 4) \sin (\frac{7 π}{4})$

$x = - 2 \sqrt{2}$

$y = (- 4) \sin (\frac{7 π}{4})$

Step 6

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.

$x = - 2 \sqrt{2}$

$y = - 4 (- \sin (\frac{π}{4}))$

Step 7

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

$x = - 2 \sqrt{2}$

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

Step 8

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1

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

$x = - 2 \sqrt{2}$

$y = - 4 \frac{- \sqrt{2}}{2}$

Step 8.2

Factor $2$ out of $- 4$ .

$x = - 2 \sqrt{2}$

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

Step 8.3

Cancel the common factor.

$x = - 2 \sqrt{2}$

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

Step 8.4

Rewrite the expression.

$x = - 2 \sqrt{2}$

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

$x = - 2 \sqrt{2}$

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

Step 9

Multiply $- 1$ by $- 2$ .

$x = - 2 \sqrt{2}$

$y = 2 \sqrt{2}$

Step 10

The rectangular representation of the polar point $(- 4, \frac{7 π}{4})$ is $(- 2 \sqrt{2}, 2 \sqrt{2})$ .

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