Find the Exact Value arccos(cos((11pi)/4))

Problem

arccos(cos((11*π)/4))

Solution

1. Identify the range of the principal inverse cosine function, which is [0,π]

2. Determine if the input angle (11*π)/4 falls within the interval [0,π] Since (11*π)/4>π we must find a coterminal angle or use the periodicity of the cosine function.

3. Subtract multiples of 2*π to find a coterminal angle.

(11*π)/4−2*π=(11*π)/4−(8*π)/4

(11*π)/4−2*π=(3*π)/4

4. Evaluate the cosine of the angle. Since cos((11*π)/4)=cos((3*π)/4) the expression simplifies.

cos((11*π)/4)=−√(,2)/2

5. Apply the inverse cosine function. We need to find the angle θ in [0,π] such that cos(θ)=−√(,2)/2

arccos(−√(,2)/2)=(3*π)/4

Final Answer

arccos(cos((11*π)/4))=(3*π)/4

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