Subjects trigonometry

Sin Cos Shifts F449C9

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1. The problem asks to find the equations of two sinusoidal functions, one based on $y=\sin(x)$ and the other on $y=\cos(x)$, each translated horizontally (right shift) and vertically. 2. The general form for a horizontally and vertically shifted sine or cosine function is: $$y = \sin(x - h) + k$$ $$y = \cos(x - h) + k$$ where $h$ is the horizontal shift (right if positive) and $k$ is the vertical shift. 3. To find $h$ and $k$, observe the graph's key points such as peaks, troughs, and midline shifts. 4. For the sine function, identify the phase shift $h$ by locating the first peak or zero crossing closest to the origin and compare it to the standard sine wave. 5. For the cosine function, do the same by identifying the horizontal shift of its peak relative to $x=0$. 6. Determine the vertical shift $k$ by finding the midline of each wave (average of max and min values). 7. Round $h$ and $k$ to two decimal places as requested. 8. Write the final equations: $$y = \sin\bigl(x - h_{\sin}\bigr) + k_{\sin}$$ $$y = \cos\bigl(x - h_{\cos}\bigr) + k_{\cos}$$ Since the exact graph values are not provided, the user should measure the shifts from the graph and plug in the values accordingly. This method allows expressing the given sinusoidal waves as translations of $y=\sin(x)$ and $y=\cos(x)$ with the closest right shift.