Trigonometry Examples

(1.5, - \frac{7 π}{6})

$(1.5, - \frac{7 π}{6})$

Step 1

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

x = r c o s θ

$x = r c o s θ$

y = r s i n θ

$y = r s i n θ$

Step 2

Substitute in the known values of

r = 1.5

$r = 1.5$ and

θ = - \frac{7 π}{6}

$θ = - \frac{7 π}{6}$ into the formulas.

x = (1.5) cos (- \frac{7 π}{6})

$x = (1.5) \cos (- \frac{7 π}{6})$

y = (1.5) sin (- \frac{7 π}{6})

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 3

Add full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$x = 1.5 \cos (\frac{5 π}{6})$

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 4

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.

$x = 1.5 (- \cos (\frac{π}{6}))$

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 5

The exact value of

$\cos (\frac{π}{6})$ is

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

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

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 6

Multiply

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

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Step 6.1

Multiply

$- 1$ by

$1.5$ .

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

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 6.2

Combine

$- 1.5$ and

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

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

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 6.3

Multiply

$- 1.5$ by

$\sqrt{3}$ .

$x = \frac{- 2.59807621}{2}$

$y = (1.5) \sin (- \frac{7 π}{6})$

$x = \frac{- 2.59807621}{2}$

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 7

Divide

$- 2.59807621$ by

$2$ .

$x = - 1.2990381$

$y = (1.5) \sin (- \frac{7 π}{6})$

Step 8

Add full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$x = - 1.2990381$

$y = 1.5 \sin (\frac{5 π}{6})$

Step 9

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

$x = - 1.2990381$

$y = 1.5 \sin (\frac{π}{6})$

Step 10

The exact value of

$\sin (\frac{π}{6})$ is

$\frac{1}{2}$ .

$x = - 1.2990381$

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

Step 11

Combine

$1.5$ and

$\frac{1}{2}$ .

$x = - 1.2990381$

$y = \frac{1.5}{2}$

Step 12

Divide

$1.5$ by

$2$ .

$x = - 1.2990381$

$y = 0.75$

Step 13

The rectangular representation of the polar point

$(1.5, - \frac{7 π}{6})$ is

$(- 1.2990381, 0.75)$ .

$(- 1.2990381, 0.75)$