Can the maximum and minimum for $y=5\sin(5x+20)-2$ be found using algebra

trigonometry

Here is what I know about $y=5\sin(5x+20)-2$:

  1. General form of a sinusoidal function is; $y=A\sin(Bx-C)+D$

  2. Amplitude $\lvert A \rvert=5$

  3. Period $ \frac{2 \pi}{B}=\frac{2 \pi}{5}$

  4. Midline $D = -2$

  5. Horizontal shift $ \frac{C}{B}=\frac{-20}{5}=-4$

From this information I know the midpoint between the maximum and minimum is $-2$. When I graph this function I know the $y$ values are $3$ for the maximum and $-7$ for the minimum. Is there any way to substitute $3$ or $-7$ into the equation $y=5\sin(5x+20)-2$ to find the $x$ values?

I can use Desmos or a graphing calculator to find the $x$ values, but this is really the only way I can find the values. I also find it difficult to graph this equation by hand. I really have a hard time finding intercepts along with the maxima and minima.

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Best Answer

The $\sin t$ function takes values in $[-1,1]$. It takes the value $1$ when $t=\frac{\pi}{2}+2 k\pi, \,\,k \in \mathbb{Z}$ and it takes the value $-1$ when $t = -\frac{\pi}{2}+2k\pi, \,\,k \in \mathbb{Z}$. So, you function attains a maximum value of $3$ when $5x+20 =\frac{\pi}{2}+2 k\pi, \,\,k \in \mathbb{Z}$, and a minimum value of $-7$ when $5x+20 =-\frac{\pi}{2}+2 k\pi, \,\,k \in \mathbb{Z}$.

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