Chapter 4, Problem 88RE

Solution

To evaluate: The expression.

Evaluate expression $\frac{\pi }{7}$

Given information: Function ${\cos }^{-1}\left(\cos \frac{\pi }{3}\right)$

Calculation: Recall that since the sine, cosine, and tangent functions are periodic, their inverse functions only represent a specific branch of the original function.

Their ranges are presented to the left. When it before seemed that an equation such as $\cos x=1$ had an infinite number of answers, these range restrictions on the arc functions make it so the equation across $1=x$ has only one solution.

Ranges of the inverse trigonometric function:

• Arcsine: $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
• Arccosine: $\left[0,\pi \right]$
• Arctangent: $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

This is a special case as we cannot simply evaluate the value of $\cos \left(-\frac{\pi }{7}\right)$ easily without a calculator.

Let $x=\cos \left(-\frac{\pi }{7}\right)$

And also $x=\cos \left(-\frac{\pi }{7}\right)$ therefore, we can reason that the expression is equal to $\frac{\pi }{7}$ since when, the angle is in first quadrant, the output will be the same as the input even when you consider the range restrictions.