Final answer:
The exact value of cos⁻¹(cos(4pi/3)) is 2pi/3, as we are looking for the angle in the range of the inverse cosine function (0 to π) that has a cosine value of -1/2, which corresponds to 4pi/3 in the third quadrant.
Step-by-step explanation:
The student asked for the exact value of the expression cos⁻¹(cos(4pi/3)). To find this value, understand that cos⁻¹ represents the inverse cosine function, which returns the angle whose cosine is the given number. The cosine of 4pi/3, which is in the third quadrant where cosine is negative, equals -1/2.
However, the range of the arccosine (or inverse cosine) function is from 0 to π, which corresponds to angles in the first and second quadrants only.
Therefore, when we apply the inverse cosine function to cos(4pi/3), we are looking for the angle in the first or second quadrant that has the same cosine value as 4pi/3. Since cosine is positive in the first quadrant and negative in the second, we find that the angle with cosine -1/2 in that range is 2pi/3. Hence, the exact value of cos⁻¹(cos(4pi/3)) is 2pi/3.
he expression given is cos⁻¹(cos(4π/3)). In order to find the exact value of this expression, we need to understand the inverse cosine function and how it relates to the cosine function.
The inverse cosine function, cos⁻¹(x), returns the angle whose cosine is x. In this case, we need to find the angle whose cosine is cos(4π/3).
Since cos(4π/3) is equal to -1/2, the inverse cosine of -1/2 is 120°.