Question

Make (35/22)π from four π.

π π π π= (35/22)π. ​

Answer 1

Answer:

Step-by-step explanation:

Start by observing that sin 35° is close tosin 30°, which is 1/2. So we immediately know it is roughly 1/2. That's within about 7% of the actual value.

Let's try to get a better estimate.  By the angle addition identity, 

sin 35° = sin 30° cos 5° + sin 5° cos 30° = (1/2) cos 5° + sin 5° 3–√/23/2.

Now, since 5° = π/36 is a relatively small angle, we can use the approximations sin x ≈ x and cos x ≈ 1. So 

sin 35° ≈ 1/2 + (π/36)(3–√/23/2).  

Now π ≈ 22/7, and 3–√3  ≈ 7/4 because 49/16 ≈ 3. So we get

sin 35° ≈ 1/2 + (22/7)(1/36)(1/2)(7/4) = 1/2 + 11/144 = 83/144, 

This differs from the true value by less 1%.  

To get an even better estimate, one can take higher order terms in the expansions  for sin and cos, as well as the square root, and use a better approximation to π. That takes more work of course.

Another approach is calculating it using the Taylor series expansion of sin x. 

First note that 35°=2π(35/360) = π(7/36) ≈ (22/7)(7/36) = 22/36 = 11/18.

sin x ≈ x−x3/3!x−x3/3! ≈ 11/18−(11/18)3/611/18−(11/18)3/6. This is accurate to better than 0.1%, but harder to calculate by hand than 83/144.

Hope this will be help

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Answer 2

Answer:

Step-by-step explanation:

Syedkaif2Ambitious

Start by observing that sin 35° is close tosin 30°, which is 1/2. So we immediately know it is roughly 1/2. That's within about 7% of the actual value.

Let's try to get a better estimate.  By the angle addition identity, 

sin 35° = sin 30° cos 5° + sin 5° cos 30° = (1/2) cos 5° + sin 5° 3–√/23/2.

Now, since 5° = π/36 is a relatively small angle, we can use the approximations sin x ≈ x and cos x ≈ 1. So 

sin 35° ≈ 1/2 + (π/36)(3–√/23/2).  

Now π ≈ 22/7, and 3–√3  ≈ 7/4 because 49/16 ≈ 3. So we get

sin 35° ≈ 1/2 + (22/7)(1/36)(1/2)(7/4) = 1/2 + 11/144 = 83/144, 

This differs from the true value by less 1%.  

To get an even better estimate, one can take higher order terms in the expansions  for sin and cos, as well as the square root, and use a better approximation to π. That takes more work of course.

Another approach is calculating it using the Taylor series expansion of sin x. 

First note that 35°=2π(35/360) = π(7/36) ≈ (22/7)(7/36) = 22/36 = 11/18.

sin x ≈ x−x3/3!x−x3/3! ≈ 11/18−(11/18)3/611/18−(11/18)3/6. This is accurate to better than 0.1%, but harder to calculate by hand than 83/144.

Hope this will be help you