Prove that sin5 theta= 5sintheta -20sin³theta+16 sin⁵theta

Step-by-step explanation:

Express sin5θ in terms of sinθ and hence find the value of sin36

∘

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By De-moivre's Theorem we know that

(cos θ + i sin θ)

5

= cos 5θ + i sin 5θ

L.H.S. on expansion by binomial theorem is

cos

5

θ+5cos

4

θ(isinθ)+10cos

3

θ(isinθ)

2

+10cos

2

θ(isinθ)

3

+5cosθ(isinθ)

4

+(isinθ)

5

Now equate real and imaginary parts and change

sin

2

θto1−cos

2

θandcos

2

θto1−sin

2

θ

depending on the answer.

∴ cos 5θ = cos θ (16 cos

4

θ - 20 cos

2

θ + 5)

sin = 5θ = sin θ (16 sin

4

θ - 20 sin

2

θ + 5)

Deduction : If θ = 36

∘

, then 5θ = 180

∘

∴ sin 5θ = 0

Also sin 36

∘

< sin 45

∘

or sin

2

36

∘

<

2

1

Now from (2), we get

0 = s(16 s

4

- 20 s

2

+ 5), s = sin 36

∘



= 0

∴s

2

=

32

20±

400−320

=

16

10−2

5

(∵s

2

<

2

1

)

∴s=

5

10−2

5

answer: the answer is (b) amie should have multiplied

step-by-step explanation: given a sphere and a cylinder with same radius and height.

let, 'r' be the radius of the cylinder and sphere and 'h' be the height of the cylinder. according to the question,

r = h.

now, the volume of the sphere will be

and the volume of the cylinder will be

therefore, we have

thus, ammie should have multiplied the volume of the sphere

depending on others is critical to survival. my source is apex!

step-by-step explanation: