Question

find the following in degrees without using a calculator.
cos⁻¹(cos 43°) = ?°

Explanation:

Step1: Recall the inverse cosine property

The inverse cosine function, \(\cos^{-1}(x)\), is the inverse of the cosine function, \(\cos(x)\), with the domain of \(\cos^{-1}(x)\) being \([-1, 1]\) and the range being \([0^{\circ}, 180^{\circ}]\). One of the key properties of inverse functions is that if \(f\) and \(f^{-1}\) are inverse functions, then \(f^{-1}(f(x)) = x\) for \(x\) in the domain of \(f\) and \(f^{-1}(f(x))\) resulting in a value in the range of \(f^{-1}\).

For the cosine function \(\cos(x)\) and its inverse \(\cos^{-1}(x)\), we have the property \(\cos^{-1}(\cos(x)) = x\) when \(x\) is in the interval \([0^{\circ}, 180^{\circ}]\) (since the range of \(\cos^{-1}(x)\) is \([0^{\circ}, 180^{\circ}]\)).

Step2: Check the value of \(x\)

In the given problem, \(x = 43^{\circ}\). We need to check if \(43^{\circ}\) lies within the interval \([0^{\circ}, 180^{\circ}]\). Since \(0^{\circ}< 43^{\circ}< 180^{\circ}\), the property \(\cos^{-1}(\cos(x))=x\) holds for \(x = 43^{\circ}\).

So, \(\cos^{-1}(\cos(43^{\circ}))=43^{\circ}\)

Answer:

\(43\)