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4 years ago

Sin(7pi/6) help plz

Mathematics

1 answer:

NeTakaya4 years ago

8 0

Answer:

Step-by-step explanation:

Note that 7pi/6 is equivalent to 6pi/6 + 1pi/6, which is equivalent to pi plus pi/6, or 180 degrees plus 30 degrees. 7pi/6 is in Quadrant III; its "opposite side" is -1 and its "hypotenuse is 2." Thus, the sine of (7pi/6) is the ratio

opp side -1

------------------- = ------------ This matches Answer B.

hypotenuse 2

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We have

$w=7ye^x-\ln z\implies\begin{cases}\dfrac{\partial w}{\partial x}=7ye^x\\\\\dfrac{\partial w}{\partial y}=7e^x\\\\\dfrac{\partial w}{\partial z}=-\dfrac1z\end{cases}$

and

$\begin{cases}x=\ln(t^2+1)\\y=\tan^{-1}t\\z=e^t\end{cases}\implies\begin{cases}\dfrac{\mathrm dx}{\mathrm dt}=\dfrac{2t}{t^2+1}\\\\\dfrac{\mathrm dy}{\mathrm dt}=\dfrac1{t^2+1}\\\\\dfrac{\mathrm dz}{\mathrm dt}=e^t\end{cases}$

Putting everything together, we get

$\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{14ye^xt}{t^2+1}+\dfrac{7e^x}{t^2+1}-\dfrac{e^t}z$

$x=\ln(t^2+1)$ , so $e^x=e^{\ln(t^2+1)}=t^2+1$ , and $z=e^t$ , so $\frac{e^t}z=1$ .

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