By analyzing available empirical data, it has been...

By analyzing available empirical data, it has been determined that the body temperature of a species fluctuates according to the model $$T(t)=38-2.5 \cos \left[\frac{\pi}{6}(t-3)\right], \quad 0 \leq t \leq 24$$ where $T$ represents temperature in degrees Celsius and $t$ represents time (in hours) measured from 12: 00 P.M. (noon). Use an identity to express $T(t)$ in terms of the sine function.

By analyzing available empirical data, it has been determined that the body temperature of a species fluctuates according to the model
$$T(t)=38-2.5 \cos \left[\frac{\pi}{6}(t-3)\right], \quad 0 \leq t \leq 24$$
where $T$ represents temperature in degrees Celsius and $t$ represents time (in hours) measured from 12: 00 P.M. (noon). Use an identity to express $T(t)$ in terms of the sine function.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. One important identity for this context is the conversion identity between cosine and sine, which states that cos(?) can be expressed as sin(? + ?/2). This identity is essential when you need to express a cosine function in terms of sine, allowing you to adjust the function’s phase accordingly.

Phase Shift

The phase shift represents a horizontal translation of the graph of a trigonometric function. When converting between sine and cosine, a phase shift is typically applied to ensure that the transformed function accurately mirrors the behavior of the original function. Understanding phase shifts is crucial for correctly aligning the peaks, troughs, and zeros of the functions after the transformation.

Periodic Functions

Periodic functions, such as sine and cosine, repeat their values in regular intervals, making them well-suited for modeling cyclic phenomena like temperature fluctuations. Their properties allow them to accurately represent processes that exhibit regular repetition over time, which is a fundamental aspect in empirical modeling where periodic behavior is observed.

Transcript

Please give Ace some feedback