Find the Exact Value arccos(sin((7pi)/4))

Problem

arccos(sin((7*π)/4))

Solution

1. Evaluate the inner trigonometric function sin((7*π)/4) The angle (7*π)/4 is in the fourth quadrant, where the sine value is negative.

sin((7*π)/4)=−√(,2)/2

2. Substitute this value back into the inverse cosine function.

arccos(−√(,2)/2)

3. Determine the angle θ such that cos(θ)=−√(,2)/2 within the restricted range of the arccosine function, which is [0,π]

cos(θ)=−√(,2)/2

4. Identify the specific angle in the second quadrant that satisfies this condition.

θ=(3*π)/4

Final Answer

arccos(sin((7*π)/4))=(3*π)/4

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