Find the exact value of the following expression: [tex]$\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)$[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. [tex]$\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)=[/tex] $\square$ (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed.)
Answer (2)
The problem asks for the exact value of cos − 1 ( 2 2 ) .
We need to find an angle θ in the range [ 0 , π ] such that cos ( θ ) = 2 2 .
Recall that cos ( 4 π ) = 2 2 .
Therefore, cos − 1 ( 2 2 ) = 4 π .
Explanation
Understanding the Problem We are asked to find the exact value of cos − 1 ( 2 2 ) . This means we need to find the angle θ such that cos ( θ ) = 2 2 , and 0 ≤ θ ≤ π .
Finding the Angle Recall the unit circle and the values of cosine for common angles. We know that cos ( 4 π ) = 2 2 . Since 4 π is in the range [ 0 , π ] , it is the correct answer.
Final Answer Therefore, cos − 1 ( 2 2 ) = 4 π .
Examples
Imagine you are designing a satellite dish. The optimal angle of the dish depends on the inverse cosine of certain ratios related to the signal strength. Knowing the exact value of inverse trigonometric functions helps you precisely align the dish to maximize signal reception.
The exact value of cos − 1 ( 2 2 ) is 4 π . This angle corresponds to 45 degrees, where the cosine value equals 2 2 . Thus, the correct answer is 4 π .
;
