What is sin⁡(2arcsin⁡(35)) ?

Question

Frauental91 · Accepted Answer

arcsin⁡(35) is some θ between −π2 and π2 with sin⁡θ=35Furthermore, with −π2≤θ≤π2 and sin⁡θ a positive number, we conclude that θ is between 0 and π2.We want to find sin⁡2θ and we already know sin⁡θ. So if we find cos⁡θ, then we can ue the double angle formula for sine.You&#039;ve probably done this kind of problem many times by now. θ is in the first quadrant and sin⁡θ=35, find cos⁡θ.Use your favorite method -- draw a triangle, or a unit circle, or an angle in standard position, or skip the picture and use cos⁡θ=±1−sin2θ (recall that our θ is in Quadrant 1, so its cosine is positive.)All of the above is really explanation of our thought process.All we really need to write is something like:Let θ=arcsin⁡(35), then sin⁡θ=35 andcos⁡θ=45And sin⁡(2θ)=2sin⁡θcos⁡θSo, putting it all together we get:sin⁡(2arcsin⁡(35))=2(35)(45)=2425

Jacob Trujillo · Accepted Answer

Evaluate arcsin⁡(35).  sin⁡(2⋅0.6435011)  Multiply 2 by 0.6435011  sin⁡(1.28700221)  The result can be shown in multiple forms.  Exact Form:  sin⁡(2(arcsin⁡(35)))  Decimal Form: 0.96

What is \sin(2\arcsin(\frac{3}{5})) ?

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