Subjects trigonometry

Sine Cosine Translation C5E441

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1. The problem asks to find the equations of the given sine and cosine graphs as translations of the basic functions $y=\sin(x)$ and $y=\cos(x)$, including horizontal shifts (phase shifts) and vertical shifts. 2. The general form for a translated sine or cosine function is: $$y = \sin(x - c) + d$$ $$y = \cos(x - c) + d$$ where $c$ is the horizontal shift (right shift if positive) and $d$ is the vertical shift. 3. To find $c$ and $d$, identify the horizontal shift by locating the new position of a key point such as the sine or cosine wave's peak or zero crossing compared to the original function. 4. The vertical shift $d$ is the amount the entire graph is moved up or down from the original $y=0$ axis. 5. Since the graph is not provided here, assume the closest right shift $c$ and vertical shift $d$ are given or measured from the graph. 6. Round the values of $c$ and $d$ to two decimal places. 7. The final equations are: $$y = \sin(x - c) + d$$ $$y = \cos(x - c) + d$$ Replace $c$ and $d$ with the measured values from the graph. This completes the translation of the sine and cosine functions based on the graph's shifts.