Trigonometry Examples

$(2, \frac{7 π}{4})$

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 $r = 2$ and $θ = \frac{7 π}{4}$ into the formulas.

$x = (2) \cos (\frac{7 π}{4})$

$y = (2) \sin (\frac{7 π}{4})$

Step 3

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

$x = 2 \cos (\frac{π}{4})$

$y = (2) \sin (\frac{7 π}{4})$

Step 4

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

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

$y = (2) \sin (\frac{7 π}{4})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

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

$y = (2) \sin (\frac{7 π}{4})$

Step 5.2

Rewrite the expression.

$x = \sqrt{2}$

$y = (2) \sin (\frac{7 π}{4})$

$x = \sqrt{2}$

$y = (2) \sin (\frac{7 π}{4})$

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.

$x = \sqrt{2}$

$y = 2 (- \sin (\frac{π}{4}))$

Step 7

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

$x = \sqrt{2}$

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

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

$x = \sqrt{2}$

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

Step 8.2

Cancel the common factor.

$x = \sqrt{2}$

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

Step 8.3

Rewrite the expression.

$x = \sqrt{2}$

$y = - \sqrt{2}$

$x = \sqrt{2}$

$y = - \sqrt{2}$

Step 9

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

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