Find the exact value of the following expression:
$\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
Answer (2)
The exact value of cos − 1 ( − 2 3 ) is 6 5 π , found in the second quadrant where cosine is negative. This angle corresponds to a reference angle of 6 π . Thus, the final answer is 6 5 π .
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We need to find the angle θ in the interval [ 0 , π ] such that cos ( θ ) = − 2 3 .
We know that cos ( 6 π ) = 2 3 , so the reference angle is 6 π .
Since the cosine is negative, the angle must be in the second quadrant, so θ = π − 6 π = 6 5 π .
Therefore, the exact value of the expression 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 We know that cos ( 6 π ) = 2 3 . Since the cosine function is negative in the second quadrant, we are looking for an angle in the second quadrant that has a reference angle of 6 π .
Calculating the Angle To find the angle in the second quadrant, we can subtract the reference angle from π : θ = π − 6 π = 6 6 π − 6 π = 6 5 π . Therefore, cos ( 6 5 π ) = − 2 3 .
Final Answer Since 6 5 π is in the range [ 0 , π ] , we have found the exact value of the given expression.
Examples
The inverse cosine function is used in various fields such as physics and engineering to find angles when the cosine of the angle is known. 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. This is also applicable in navigation, where angles are crucial for determining direction and position.
