Calculating sin ( 0.5 arcsinx)

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In summary, the conversation discusses using two different trig identities to solve a question and obtaining different answers when graphed on Wolfram Alpha. The first identity, sinx = 2sin(0.5x)cos(0.5x), when applied to the question of sin(0.5arcsinx), gives the answer of x/sqrt(2(1+sqrt(1-x^2))). The second identity, sin2x = 0.5(1-cos2x), when applied to the question of sin(0.5arcsinx), gives the answer of sqrt(1/2(1-sqrt(1-x^2))). Despite the different forms, these answers are equivalent due to the
  • #1
Dethrone
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I did this question using two different trig identities, each of which gave me a different answer when I graphed them on Wolfram Alpha.

1. First identity: sinx = 2sin(0.5x)cos(0.5x)
isolating for sin(0.5x):

sin(0.5x) = (sinx) / (2cos(0.5x))

The question wants sin (0.5 arcsinx):

Applying the formula, we get the answer to be
\(\displaystyle \frac{x}{\sqrt{2(1+\sqrt{1-x^2})}}\)

2. Second identity: sin2x = 0.5 (1-cos2x),
or sin20.5x = 0.5(1-cosx)

Applying that formula, we get the answer to be
\(\displaystyle \sqrt{\frac{1}{2}(1-\sqrt{1-x^2})}\)

Why are the two answers different? Is my algebra wrong somewhere, or is this an issue of domain or something?
 
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  • #2
Rido12 said:
I did this question using two different trig identities, each of which gave me a different answer when I graphed them on Wolfram Alpha.

1. First identity: sinx = 2sin(0.5x)cos(0.5x)
isolating for sin(0.5x):

sin(0.5x) = (sinx) / (2cos(0.5x))

The question wants sin (0.5 arcsinx):

Applying the formula, we get the answer to be
\(\displaystyle \frac{x}{\sqrt{2(1+\sqrt{1-x^2})}}\)

2. Second identity: sin2x = 0.5 (1-cos2x),
or sin20.5x = 0.5(1-cosx)

Applying that formula, we get the answer to be
\(\displaystyle \sqrt{\frac{1}{2}(1-\sqrt{1-x^2})}\)

Why are the two answers different? Is my algebra wrong somewhere, or is this an issue of domain or something?
They are the same
\(\displaystyle \displaystyle \frac{x}{\sqrt{2(1+\sqrt{1-x^2})}}= \frac{x \sqrt{1 -\sqrt{ 1-x^2}}}{\sqrt{2(1+\sqrt{1-x^2})}\sqrt{1 - \sqrt{1-x^2}}}= \frac{ x \sqrt{ 1 - \sqrt{ 1 -x^2 }} }{ \sqrt{ 2 ( 1 - (1 - x^2 ))} }= \sqrt{ \frac{1}{2} ( 1 - \sqrt{1 - x^2 } ) } \)

Note that $ \sqrt{x^2} = \mid x \mid $
 
  • #4
It has been my experience that W|A does not handle imaginary numbers correctly with respect to graphing or odd roots of real negative values. Try graphing \(\displaystyle y=x^{\frac{1}{3}}\) and you get garbage for negative values of $x$.
 
  • #5
Above, I used sin20.5x = 0.5(1-cosx), but isolating \(\displaystyle \sin\left({x}\right) \)would give us |\(\displaystyle \sin\left({x}\right)\)|. And |\(\displaystyle \sin\left({x}\right)\)| is only equal to \(\displaystyle \sin\left({x}\right)\) on intervals such as \(\displaystyle (0, \pi)\) and \(\displaystyle (2\pi, 3\pi)\). Why, then, do they yield the same answer?
 

FAQ: Calculating sin ( 0.5 arcsinx)

What is the formula for calculating sin ( 0.5 arcsinx)?

The formula for calculating sin ( 0.5 arcsinx) is sin ( 0.5 arcsinx) = √((1 - cos( arcsinx))/2).

How do I find the value of sin ( 0.5 arcsinx) using a calculator?

To find the value of sin ( 0.5 arcsinx) using a calculator, first input the value of arcsinx into the calculator. Then, take the cosine of that value and subtract it from 1. Finally, take the square root of that value and divide it by 2 to get the final result.

Can I calculate sin ( 0.5 arcsinx) without using a calculator?

Yes, you can calculate sin ( 0.5 arcsinx) without using a calculator by using the formula sin ( 0.5 arcsinx) = √((1 - cos( arcsinx))/2). However, this may be more time-consuming and prone to errors compared to using a calculator.

What is the range of values for sin ( 0.5 arcsinx)?

The range of values for sin ( 0.5 arcsinx) is between -1 and 1, inclusive.

What is the inverse of calculating sin ( 0.5 arcsinx)?

The inverse of calculating sin ( 0.5 arcsinx) is finding the value of arcsinx given the value of sin ( 0.5 arcsinx). This can be done by rearranging the formula sin ( 0.5 arcsinx) = √((1 - cos( arcsinx))/2) to solve for arcsinx.

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