Suppose that r (cos(θ) + i sin(θ)) is a third root of 27 (cos(π/5) + i sin(π/5)), i.e.
[r (cos(θ) + i sin(θ))]³ = 27 (cos(π/5) + i sin(π/5))
Expand the left side using de Moivre's theorem:
r³ (cos(3θ) + i sin(3θ)) = 27 (cos(π/5) + i sin(π/5))
Matching up real and imaginary parts, we have
r³ cos(3θ) = 27 cos(π/5)
r³ sin(3θ) = 27 sin(π/5)
Right away we see that r³ = 27 ⇒ r = 3, which eliminates A.
We also have
(r³ sin(3θ)) / (r³ cos(3θ)) = (27 sin(π/5)) / (27 cos(π/5))
tan(3θ) = tan(π/5)
3θ = tan⁻¹(tan(π/5))
3θ = π/5 + nπ
(where n is an integer)
θ = π/15 + nπ/3
Now,
• n = 0 ⇒ θ = π/15 ⇒ D is a third root
• n = 1 ⇒ θ = π/15 + π/3 = 2π/5
• n = 2 ⇒ θ = π/15 + 2π/3 = 11π/15 ⇒ C is a third root
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Answer:
y = 4x + 3
Step-by-step explanation:
The output of 3 at x = 0 tells you the y-intercept is 3.
The change of 4 from 3 to 7 for an input change of 1 from 0 to 1 tells you the slope is 4/1 = 4. Then the slope-intercept form of the equation is ...
y = mx + b . . . . . m = slope, b = y-intercept
y = 4x + 3
_____
It is pretty conventional to use y for the output variable and x for the input variable. Your rule could be ...
output = 4 × input + 3
Given:
convex 41-gon
n = 41
Required:
Sum of interior angles.
Solution:
To solve for the sum of interior angles, we can use the formula:
Sum of interior angles = 180(n-2)°
Substituting,
Sum of interior angles = 180(41-2)°
Sum of interior angles = 7020°
Therefore, the sum of the measures of the interior angle of a convex 41-gon is 7020°.
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