What is the Cartesian form of $(- 64, \frac{5 π}{12})$ $(-64,(5pi)/12)$ ?

Question

What is the Cartesian form of $(- 64, \frac{5 π}{12})$ $(-64,(5pi)/12)$ ?

1 Answer

Impact of this question

1510 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-22T15:09:00

I got $(- 16 \sqrt{6} + 16 \sqrt{2}, - 16 \sqrt{6} - 16 \sqrt{2})$ $(-16sqrt6 + 16sqrt2","-16sqrt6 - 16sqrt2)$ , and Wolfram Alpha gives the same answer as well:

Since you have two coordinates, $(r, θ)$ $(r,theta)$ , you are in polar coordinates, a radial direction and a single angle. You can tell because the second coordinate is an angle in radians.

(If you had three coordinates, you would have to specify spherical or cylindrical coordinates.)

In polar coordinates, recall that

$x = r cos θ$ $x = rcostheta$ ,
$y = r sin θ$ $y = rsintheta$ .

Therefore, you input the values of $r$ $r$ and $θ$ $theta$ such that you get your new $(x, y)$ $(x,y)$ coordinates.

$(x$ $color(blue)("("x)$ $,$ $color(blue)(,)$ $y)$ $color(blue)(y")")$

$= (r cos θ, r sin θ)$ $= (rcostheta,rsintheta)$

$= (- 64 cos (\frac{5 π}{12}), - 64 sin (\frac{5 π}{12}))$ $= (-64cos((5pi)/12),-64sin((5pi)/12))$

$= (- 64 cos (75^{\circ}), - 64 sin (75^{\circ}))$ $= (-64cos(75^@),-64sin(75^@))$

Now we can use the additive angle formulas

$sin (u + v) = sin u sin v + cos u cos v$ $sin(u+v) = sinusinv + cosucosv$ ,
$cos (u + v) = cos u cos v - sin u sin v$ $cos(u+v) = cosucosv - sinusinv$ ,

to get

$= (- 64 cos (30^{\circ} + 45^{\circ}), - 64 sin (30^{\circ} + 45^{\circ}))$ $= (-64cos(30^@+45^@),-64sin(30^@+45^@))$

$= (- 64 ({cos 30}^{\circ} {cos 45}^{\circ} - {sin 30}^{\circ} {sin 45}^{\circ}), - 64 ({sin 30}^{\circ} {cos 45}^{\circ} + {cos 30}^{\circ} {sin 45}^{\circ}))$ $= (-64(cos30^@cos45^@ - sin30^@sin45^@),-64(sin30^@cos45^@ + cos30^@sin45^@))$

$= (- 64 (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}), - 64 (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}))$ $= (-64(sqrt3/2*sqrt2/2 - 1/2*sqrt2/2),-64(1/2*sqrt2/2 + sqrt3/2*sqrt2/2))$

$= (- 64 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}), - 64 (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}))$ $= (-64(sqrt6/4 - sqrt2/4),-64(sqrt2/4 + sqrt6/4))$

$= (- 64 (\frac{\sqrt{6} - \sqrt{2}}{4}), - 64 (\frac{\sqrt{6} + \sqrt{2}}{4}))$ $= (-64((sqrt6-sqrt2)/4),-64((sqrt6+sqrt2)/4))$

$= (- 16 (\sqrt{6} - \sqrt{2}), - 16 (\sqrt{6} + \sqrt{2}))$ $= (-16(sqrt6-sqrt2),-16(sqrt6+sqrt2))$

$= (- 16 \sqrt{6} + 16 \sqrt{2}, - 16 \sqrt{6} - 16 \sqrt{2})$ $= color(blue)((-16sqrt6 + 16sqrt2","-16sqrt6 - 16sqrt2))$ .

Science

Math

Humanities

... and beyond

What is the Cartesian form of $(- 64, \frac{5 π}{12})$ $(-64,(5pi)/12)$ ?

1 Answer

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

What is the Cartesian form of (-64,(5pi)/12)(−64,5π12)(-64,(5pi)/12)?

1 Answer

Related questions

Impact of this question

What is the Cartesian form of $(- 64, \frac{5 π}{12})$ $(-64,(5pi)/12)$ ?