n the month of June, the temperature in Johannesburg, South Africa, varies over the day in a periodic way that can be modeled approximately by a trigonometric function.

The trigonometric functions are actual functions that relate a right-angled triangle's angle to side length ratios. They are extensively employed in all geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many more.

According to the given information :

Maximum temp. = 18 degree Celsius

Minimum temp. = 3 degree Celsius

The half way between 10 am and 10 pm is 4 am

Temperature data changes throughout the year can be modeled using the sine and cosine functions. The following form of equation can be used to model these data:

T = A cos B(t - C) + D, where A,B,C,D, are constants, T is the temperature in °C and t is the hour (1–24)

A = amplitude = (Tmax - Tmin)/2

A = (3 - 18)/2 = - 15/2 = -7.5

B = 2π/24 = π/12

C = units translated to the right

C = 4

D = ymin + amplitude = units translated up

D = 7.5 + 3 = 10.5

The formula of the trigonometric function that models the temperature T in Johannesburg t hours after midnight we be

T = - 7.5 Cos π/12( t - 4 ) + 10.5

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