How to Approximate arccos(1/4) by Hand?

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In summary, arccosine, also known as inverse cosine, is a mathematical function that gives the angle whose cosine is a given number. The approximation of arccos(1/4) is approximately 1.3181 radians or 75.5224 degrees and is typically calculated using a mathematical formula or a calculator. This approximation is useful in science for solving problems involving angles and cosine functions, and has real-world applications in engineering, physics, and astronomy.
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how to aproximate arccos(1/4) by hand?
 
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Well the Taylor Series for arccos(x) begins as pi/2 - x - x^3/6, since (1/4)^3/6 = .003 this is a pretty good approximation:

[tex]arccos(1/4) \approx \pi/2 - x - x^{3}/3 \approx 1.318[/tex]

knowing that pi/2 is about 1.57 might help too.
 
  • #3
The derivative of f(x) = arccos(x) can be written as a power series (a binomial series). Find this series, integrate both sides, check for convergence, and plug in x = 1/4.
 

FAQ: How to Approximate arccos(1/4) by Hand?

What is the definition of arccosine?

Arccosine, also known as inverse cosine, is a mathematical function that gives the angle whose cosine is a given number. It is the inverse function of the cosine function.

What is the approximation of arccos(1/4)?

The approximation of arccos(1/4) is approximately 1.3181 radians or 75.5224 degrees. This can also be written as arccos(0.25).

How is the approximation of arccos(1/4) calculated?

The approximation of arccos(1/4) is typically calculated using a mathematical formula or a calculator. The formula for calculating arccos(x) is arccos(x) = cos-1(x). In this case, x is equal to 1/4, giving us arccos(1/4) = cos-1(1/4).

Why is the approximation of arccos(1/4) useful in science?

The approximation of arccos(1/4) is useful in science because it helps solve problems involving angles and cosine functions. It can also be used to find the inverse of a cosine function, which is used in many scientific calculations.

What are some real-world applications of the approximation of arccos(1/4)?

The approximation of arccos(1/4) has many real-world applications, including in engineering, physics, and astronomy. It can be used to calculate the angles of a triangle, determine the phase difference in AC circuits, and find the position of planets and stars in the night sky.

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