The height, [tex]$d$[/tex], in feet of a ball suspended from a spring as a function of time, [tex]$t$[/tex], in seconds can be modeled by the equation [tex]$d=-2 \sin \left(\pi\left(t+\frac{1}{2}\right)\right)+5$[/tex]. The ball is released from its lowest point at [tex]$t=0$[/tex] seconds. Using your knowledge of the general form of sine and cosine functions, which of the following equations can also model this situation?
[tex]$d=-2 \cos (\pi t)+5$[/tex]
[tex]$d=-2 \cos \left(\pi\left(t+\frac{1}{2}\right)\right)+5$[/tex]
[tex]$d=2 \cos (\pi t)+5$[/tex]
[tex]$d=2 \cos \left(\pi\left(t+\frac{1}{2}\right)\right)+5$[/tex]
Answer (1)
Rewrite the given equation: d = − 2 sin ( π t + 2 π ) + 5 .
Apply the trigonometric identity sin ( x + 2 π ) = cos ( x ) : d = − 2 cos ( π t ) + 5 .
The equivalent equation is d = − 2 cos ( π t ) + 5 .
Therefore, the answer is d = − 2 cos ( π t ) + 5 .
Explanation
Using Trigonometric Identity We are given the equation d = − 2 sin ( π ( t + 2 1 ) ) + 5 and we want to find an equivalent cosine function. We can use the trigonometric identity sin ( x + 2 π ) = cos ( x ) .
Rewriting the Equation First, rewrite the given equation as: d = − 2 sin ( π t + 2 π ) + 5
Applying the Identity Now, apply the identity sin ( x + 2 π ) = cos ( x ) where x = π t :
d = − 2 cos ( π t ) + 5
Finding the Equivalent Equation Comparing this result with the given options, we find that the equivalent equation is d = − 2 cos ( π t ) + 5 .
Examples
Understanding sinusoidal functions is crucial in many fields. For example, in electrical engineering, alternating current (AC) can be modeled using sine or cosine functions. If you have a circuit with a voltage that varies sinusoidally with time, you can use these functions to analyze the behavior of the circuit. Similarly, in acoustics, sound waves can be modeled using sinusoidal functions, allowing engineers to design better audio equipment and analyze sound propagation.
