Find the exact value of the following expression.
$\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
Select the correct choice below and, if necessary, fill in the answer.
A. $\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)=$ $\square$ (Simplify your answer. Type an exact answer, using $\pi$ as needed.)
B. The function is not defined.
Answer (2)
The problem asks to find the exact value of cos − 1 ( − 2 3 ) .
We need to find an angle θ in the interval [ 0 , π ] such that cos ( θ ) = − 2 3 .
The angle is θ = 6 5 π .
Therefore, the exact value is 6 5 π .
Explanation
Understanding the problem We are asked to find the exact value of cos − 1 ( − 2 3 ) . This means we need to find an angle θ such that cos ( θ ) = − 2 3 , and θ is in the range of the inverse cosine function, which is [ 0 , π ] .
Finding the reference angle Recall the unit circle and the values of cosine for common angles. We know that cos ( 6 π ) = 2 3 . Since we are looking for an angle whose cosine is − 2 3 , we need to find an angle in the second quadrant (where cosine is negative) that has a reference angle of 6 π .
Determining the angle The angle in the second quadrant with a reference angle of 6 π is π − 6 π = 6 5 π . Therefore, cos ( 6 5 π ) = − 2 3 .
Final Answer Since 6 5 π is in the range of the inverse cosine function (i.e., [ 0 , π ] ), we have cos − 1 ( − 2 3 ) = 6 5 π .
Examples
The inverse cosine function is used in various fields such as physics and engineering to find angles. For example, if you know the ratio of adjacent side to hypotenuse of a right triangle is − 2 3 , you can use the inverse cosine function to find the angle. In navigation, if you know the direction cosine between two points, you can use the inverse cosine to find the angle between the directions. This is also used in computer graphics to calculate angles for rotations and projections.
The exact value of cos − 1 ( − 2 3 ) is 6 5 π . This angle is in the range [ 0 , π ] where the cosine function yields a negative value. Thus, the answer is 6 5 π .
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