Principal Value of cos^-1(cos 5)

To find the principal value of cos^-1(cos 5), we need to understand the concept of inverse trigonometric functions and their ranges.

Inverse Trigonometric Functions
Inverse trigonometric functions are used to find the angle or value of a trigonometric function when the value of the trigonometric function is given. For example, the inverse cosine function, cos^-1(x), gives the angle whose cosine is x.

Range of Inverse Cosine Function
The range of the inverse cosine function, cos^-1(x), is from 0 to π, or [0, π]. This means that the value of cos^-1(x) will always be between 0 and π.

Finding cos^-1(cos 5)
In this question, we are given cos 5 as the input to the inverse cosine function. Since the range of cos^-1(x) is from 0 to π, we need to find an angle between 0 and π whose cosine is cos 5.

Reducing the Angle
To find the angle whose cosine is cos 5, we can use the concept of periodicity of the cosine function. The cosine function repeats itself every 2π radians. Therefore, we can subtract or add a multiple of 2π to the given angle to find an angle between 0 and 2π whose cosine is cos 5.

Calculating the Principal Value
To find the principal value, we need to choose the angle between 0 and 2π that is closest to the given angle. In this case, the given angle is 5, and the closest angle between 0 and 2π is 2π - 5.

Therefore, the principal value of cos^-1(cos 5) is 2π - 5.