Convert to Rectangular Coordinates (1.5,-(7pi)/6)

Convert to Rectangular Coordinates (1.5,-(7pi)/6)

$(1.5, - \frac{7 π}{6})$

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

$x = r c o s θ$

$y = r s i n θ$

Substitute in the known values of $r = 1.5$ and $θ = - \frac{7 π}{6}$ into the formulas.

$x = (1.5) \cos (- \frac{7 π}{6})$

$y = (1.5) \sin (- \frac{7 π}{6})$

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$x = 1.5 \cos (\frac{5 π}{6})$

$y = (1.5) \sin (- \frac{7 π}{6})$

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

$x = 1.5 (- \cos (\frac{π}{6}))$

$y = (1.5) \sin (- \frac{7 π}{6})$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$x = 1.5 (- \frac{\sqrt{3}}{2})$

$y = (1.5) \sin (- \frac{7 π}{6})$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$x = 1.5 (\frac{- \sqrt{3}}{2})$

$y = (1.5) \sin (- \frac{7 π}{6})$

Factor $2$ out of $1.5$ .

$x = 2 (0.75) (\frac{- \sqrt{3}}{2})$

$y = (1.5) \sin (- \frac{7 π}{6})$

Cancel the common factor.

$x = 2 \cdot (0.75 (\frac{- \sqrt{3}}{2}))$

$y = (1.5) \sin (- \frac{7 π}{6})$

Rewrite the expression.

$x = 0.75 (- \sqrt{3})$

$y = (1.5) \sin (- \frac{7 π}{6})$

$x = 0.75 (- \sqrt{3})$

$y = (1.5) \sin (- \frac{7 π}{6})$

Multiply $- 1$ by $0.75$ .

$x = - 0.75 \sqrt{3}$

$y = (1.5) \sin (- \frac{7 π}{6})$

Multiply $- 0.75$ by $\sqrt{3}$ .

$x = - 1.2990381$

$y = (1.5) \sin (- \frac{7 π}{6})$

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$x = - 1.2990381$

$y = 1.5 \sin (\frac{5 π}{6})$

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

$x = - 1.2990381$

$y = 1.5 \sin (\frac{π}{6})$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$x = - 1.2990381$

$y = 1.5 (\frac{1}{2})$

Cancel the common factor of $2$ .

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Factor $2$ out of $1.5$ .

$x = - 1.2990381$

$y = 2 (0.75) (\frac{1}{2})$

Cancel the common factor.

$x = - 1.2990381$

$y = 2 \cdot (0.75 (\frac{1}{2}))$

Rewrite the expression.

$x = - 1.2990381$

$y = 0.75$

$x = - 1.2990381$

$y = 0.75$

The rectangular representation of the polar point $(1.5, - \frac{7 π}{6})$ is $(- 1.2990381, 0.75)$ .

$(- 1.2990381, 0.75)$

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