Trigonometry Examples

$(5, \frac{3 π}{4})$

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 $r = 5$ and $θ = \frac{3 π}{4}$ into the formulas.

$x = (5) \cos (\frac{3 π}{4})$

$y = (5) \sin (\frac{3 π}{4})$

Step 3

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.

$x = 5 (- \cos (\frac{π}{4}))$

$y = (5) \sin (\frac{3 π}{4})$

Step 4

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

$x = 5 (- \frac{\sqrt{2}}{2})$

$y = (5) \sin (\frac{3 π}{4})$

Step 5

Multiply $5 (- \frac{\sqrt{2}}{2})$ .

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Step 5.1

Multiply $- 1$ by $5$ .

$x = - 5 \frac{\sqrt{2}}{2}$

$y = (5) \sin (\frac{3 π}{4})$

Step 5.2

Combine $- 5$ and $\frac{\sqrt{2}}{2}$ .

$x = \frac{- 5 \sqrt{2}}{2}$

$y = (5) \sin (\frac{3 π}{4})$

$x = \frac{- 5 \sqrt{2}}{2}$

$y = (5) \sin (\frac{3 π}{4})$

Step 6

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{2}}{2}$

$y = (5) \sin (\frac{3 π}{4})$

Step 7

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

$x = - \frac{5 \sqrt{2}}{2}$

$y = 5 \sin (\frac{π}{4})$

Step 8

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

$x = - \frac{5 \sqrt{2}}{2}$

$y = 5 (\frac{\sqrt{2}}{2})$

Step 9

Combine $5$ and $\frac{\sqrt{2}}{2}$ .

$x = - \frac{5 \sqrt{2}}{2}$

$y = \frac{5 \sqrt{2}}{2}$

Step 10

The rectangular representation of the polar point $(5, \frac{3 π}{4})$ is $(- \frac{5 \sqrt{2}}{2}, \frac{5 \sqrt{2}}{2})$ .

$(- \frac{5 \sqrt{2}}{2}, \frac{5 \sqrt{2}}{2})$