Refer to the following: By analyzing available empirical...

Refer to the following: By analyzing available empirical data, it has been determined that the body temperature of a species fluctuates according to the model $$ T(t)=37.10+1.40 \sin \left(\frac{\pi}{24} t\right) \cos \left(\frac{\pi}{24} t\right) \quad 0 \leq t \leq 24 $$ where $T$ represents temperature in degrees Celsius and $t$ represents time (in hours) measured from 12: 00 A.M. (midnight). Find the time(s) of day the body temperature is $37.28^{\circ} \mathrm{C} .$ Round to the nearest hour.

Refer to the following:
By analyzing available empirical data, it has been determined that the body temperature of a species fluctuates according to the model
$$
T(t)=37.10+1.40 \sin \left(\frac{\pi}{24} t\right) \cos \left(\frac{\pi}{24} t\right) \quad 0 \leq t \leq 24
$$
where $T$ represents temperature in degrees Celsius and $t$ represents time (in hours) measured from 12: 00 A.M. (midnight).
Find the time(s) of day the body temperature is $37.28^{\circ} \mathrm{C} .$ Round to the nearest hour.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angles corresponding to specific trigonometric values. They are crucial in solving equations that arise in various applications, providing a pathway to determine the time or phase in models involving sine or cosine functions.

Domain Considerations

Domain considerations involve restricting the solutions of an equation to a particular interval, ensuring the results make sense within the physical or practical context. When modeling natural processes, it is important to consider the appropriate range of input values, such as the hours within a day.

Sinusoidal Models

Sinusoidal models are mathematical expressions that use sine and cosine functions to represent periodic phenomena. These models capture behaviors that repeat at regular intervals and are often used to model natural cycles such as temperature fluctuations, sound waves, and circadian rhythms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are useful for simplifying expressions and solving equations, such as converting a product of sine and cosine into a single sine function using the double-angle identity.

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