An arc on a circle measures 85°. The measure of the central angle, in radians, is within which range? 0 to StartFraction pi Over
2 answers:
Answer:
0 to StartFraction pi Over 2
Step-by-step explanation:
we know that
If the arc on a circle measures 85 degrees, then the measure of the central angle is also 85 degrees too
so
The central angle in degrees is within the range
Convert to radians
Remember that
therefore
the range in radians is
0 to StartFraction pi Over 2
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By using <span>De Moivre's theorem:
</span>
If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴![\sqrt[n]{z} = \sqrt[n]{a} \ (cos \ \frac{\theta + 360K}{n} + i \ sin \ \frac{\theta +360k}{n} )](https://tex.z-dn.net/?f=%20%5Csqrt%5Bn%5D%7Bz%7D%20%3D%20%20%5Csqrt%5Bn%5D%7Ba%7D%20%5C%20%28cos%20%5C%20%20%5Cfrac%7B%5Ctheta%20%2B%20360K%7D%7Bn%7D%20%2B%20i%20%5C%20sin%20%5C%20%5Cfrac%7B%5Ctheta%20%2B360k%7D%7Bn%7D%20%29)
k= 0, 1 , 2, ..... , (n-1)
For The given complex number <span>⇒ z = 81(cos(3π/8) + i sin(3π/8))
</span>
Part (A) <span>find the modulus for all of the fourth roots
</span>
<span>∴ The modulus of the given complex number = l z l = 81
</span>
∴ The modulus of the fourth root =![\sqrt[4]{z} = \sqrt[4]{81} = 3](https://tex.z-dn.net/?f=%20%5Csqrt%5B4%5D%7Bz%7D%20%3D%20%20%5Csqrt%5B4%5D%7B81%7D%20%3D%203)
Part (b) find the angle for each of the four roots
The angle of the given complex number =
There is four roots and the angle between each root =
The angle of the first root =
The angle of the second root =
The angle of the third root =
The angle of the fourth root =
Part (C): find all of the fourth roots of this
The first root =
The second root =
The third root =
The fourth root =
</span>
If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴
k= 0, 1 , 2, ..... , (n-1)
For The given complex number <span>⇒ z = 81(cos(3π/8) + i sin(3π/8))
</span>
Part (A) <span>find the modulus for all of the fourth roots
</span>
<span>∴ The modulus of the given complex number = l z l = 81
</span>
∴ The modulus of the fourth root =
Part (b) find the angle for each of the four roots
The angle of the given complex number =
There is four roots and the angle between each root =
The angle of the first root =
The angle of the second root =
The angle of the third root =
The angle of the fourth root =
Part (C): find all of the fourth roots of this
The first root =
The second root =
The third root =
The fourth root =