Trigonometry Examples

$(7, - 30)$

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 $r = 7$ and $θ = - 30$ into the formulas.

$x = (7) \cos (- 30)$

$y = (7) \sin (- 30)$

Step 3

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

$x = 7 \cos (30)$

$y = (7) \sin (- 30)$

Step 4

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

$x = 7 (\frac{\sqrt{3}}{2})$

$y = (7) \sin (- 30)$

Step 5

Combine $7$ and $\frac{\sqrt{3}}{2}$ .

$x = \frac{7 \sqrt{3}}{2}$

$y = (7) \sin (- 30)$

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.

$x = \frac{7 \sqrt{3}}{2}$

$y = 7 (- \sin (30))$

Step 7

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

$x = \frac{7 \sqrt{3}}{2}$

$y = 7 (- \frac{1}{2})$

Step 8

Multiply $7 (- \frac{1}{2})$ .

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Step 8.1

Multiply $- 1$ by $7$ .

$x = \frac{7 \sqrt{3}}{2}$

$y = - 7 (\frac{1}{2})$

Step 8.2

Combine $- 7$ and $\frac{1}{2}$ .

$x = \frac{7 \sqrt{3}}{2}$

$y = \frac{- 7}{2}$

$x = \frac{7 \sqrt{3}}{2}$

$y = \frac{- 7}{2}$

Step 9

Move the negative in front of the fraction.

$x = \frac{7 \sqrt{3}}{2}$

$y = - \frac{7}{2}$

Step 10

The rectangular representation of the polar point $(7, - 30)$ is $(\frac{7 \sqrt{3}}{2}, - \frac{7}{2})$ .

$(\frac{7 \sqrt{3}}{2}, - \frac{7}{2})$