`arccos(1/2)` Evaluate the expression without using a calculator

Sunday, July 14, 2013

$\displaystyle{\arccos{{\left(\frac{{1}}{{2}}\right)}}}$ Evaluate the expression without using a calculator

$\displaystyle{\arccos{{\left(\frac{{1}}{{2}}\right)}}}$

Let this expression be equal to y.

$\displaystyle{y}={\arccos{{\left(\frac{{1}}{{2}}\right)}}}$

Rewriting this in terms of cosine function the equation becomes:

$\displaystyle{\cos{{\left({y}\right)}}}=\frac{{1}}{{2}}$

Base on the Unit Circle Chart, cosine is 1/2 at angles pi/3 and (5pi)/3.

$\displaystyle{y}=\frac{\pi}{{3}},\frac{{{5}\pi}}{{3}}$

Then, consider the original equation again.

$\displaystyle{y}={\arccos{{\left(\frac{{1}}{{2}}\right)}}}$

Take note that the range of arccosine is $\displaystyle{0}\leq{y}\leq\pi$ . Between $\displaystyle\frac{\pi}{{3}}$ and $\displaystyle\frac{{{5}\pi}}{{3}}$ , it is only $\displaystyle\frac{\pi}{{3}}$ that belongs to this interval. So the solution to the original equation is:

$\displaystyle{y}={\arccos{{\left(\frac{{1}}{{2}}\right)}}}$

$\displaystyle{y}=\frac{\pi}{{3}}$

Therefore, $\displaystyle{\arccos{{\left(\frac{{1}}{{2}}\right)}}}=\frac{\pi}{{3}}$ .

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Sunday, July 14, 2013

$\displaystyle{\arccos{{\left(\frac{{1}}{{2}}\right)}}}$ Evaluate the expression without using a calculator

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Sunday, July 14, 2013

Evaluate the expression without using a calculator

No comments:

Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{\arccos{{\left(\frac{{1}}{{2}}\right)}}}$ Evaluate the expression without using a calculator

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?