A parametric curve is described by x=sin⁡t ,y=sin⁡(t+π6) Is there a way to show that y=32x+1−x22 without using the property sin⁡(a+b)=sin⁡acos⁡b+sin⁡bcos⁡a?

Since sin⁡(t+π6)=32sin⁡t+12cos⁡t, we have y=32x+121−x2 Without using the formula, I feel like arcsin⁡y−π6=t=arcsin⁡x

Question: "A parametric curve is described by x=sin⁡t ,y=sin⁡(t+π6)..."

arrebolyt

2022-01-22

A parametric curve is described by

x = \sin t, y = \sin (t + \frac{π}{6})

Is there a way to show that

y = \frac{\sqrt{3}}{2} x + \frac{\sqrt{1 - x^{2}}}{2}

without using the property

\sin (a + b) = \sin a \cos b + \sin b \cos a

tainiaadjouctlw

Beginner2022-01-23Added 14 answers

Since

\sin (t + \frac{π}{6}) = \frac{3}{\sqrt{2}} \sin t + \frac{1}{2} \cos t

, we have

y = \frac{3}{\sqrt{2}} x + \frac{1}{2} \sqrt{1 - x^{2}}

Without using the formula, I feel like

\arcsin y - \frac{π}{6} = t = \arcsin x

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