Let z1 = 8(cos(5π)/(14) + i sin(5π)/(14)) and z2 =...

Let $z_1 = 8\left(\cos \frac{5\pi}{14} + i \sin \frac{5\pi}{14}\right)$ and $z_2 = 10\left(\cos \frac{10\pi}{7} + i \sin \frac{10\pi}{7}\right)$. \\ Find the following. \\ Write your answers in trigonometric form. \\ Give exact values in your answers (not decimal approximations). \\ (a) $z_1 z_2 = \boxed{}\left(\cos\boxed{} + i \sin\boxed{}\right)$ \\ (b) $\frac{z_1}{z_2} = \boxed{}\left(\cos\boxed{} + i \sin\boxed{}\right)$

Transcript

00:01 In this question given that cos -5 is equal to 0 .3031 -4 -5, 3 pi by 2 to 2 pi, and sine theta is negative 0 .3381 4 theta, pi to 3 pi by 2.

00:17 And here, by using these data, we will find the value of sine 5, 10 -5, cos theta and 10 theta.

00:25 So let's start to solve this problem.

00:27 Here this is coordinate system, this is first quadrant, this is second quadrant, this is third quadrant and this is fourth quadrant.

00:39 Here, phi belongs to fourth quadrant, so here five.

00:44 And theta belongs to third quadrant.

00:49 So here theta.

00:52 And we know in a fourth quadrant only course and secar position.

00:59 And in a third quadrant only 10 and code are positive.

01:06 Now first we find the value of sine 5 and 10 5.

01:09 So here the value of cos 5 is equal to 0 .303 and 1 and we know cos 5 is the ratio of base over hypotenous.

01:21 Here base is equal to 0 .3031 and let us assume that the hypotenous is equal to 1.

01:29 So base over hypotenus will come equal to 0 .3031.

01:34 Now by using pythagoras theorem we can find perpendicular and perpendicular is equal to square root of hypotenous square minus base square.

01:45 So perpendicular will come equal to 1 minus 0 .3031 whole square and this will come equals to 0 .9529.

01:59 Now we know sine phi is the ratio of perpendicular and hypotenous.

02:09 And here this ratio is negative because phi lies in the fourth quadrant.

02:16 Now we put the value of here p and h.

02:20 So by putting the value of p and h, we obtain the value of p and h, we obtain the value of sine 5 is equal to negative 0 .95 to 9...

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