Simplify sec((3pi)/4)

Problem

sec((3*π)/4)

Solution

1. Identify the relationship between the secant function and the cosine function using the reciprocal identity.

sec(θ)=1/cos(θ)

2. Substitute the given angle into the identity.

sec((3*π)/4)=1/cos((3*π)/4)

3. Determine the value of the cosine function at the angle (3*π)/4 Since this angle is in the second quadrant, the cosine value is negative.

cos((3*π)/4)=−√(,2)/2

4. Calculate the reciprocal of the cosine value to find the secant.

sec((3*π)/4)=1/(−√(,2)/2)

5. Simplify the fraction by multiplying by the reciprocal of the denominator.

sec((3*π)/4)=−2/√(,2)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

sec((3*π)/4)=−(2√(,2))/2

7. Reduce the expression to its simplest form.

sec((3*π)/4)=−√(,2)

Final Answer

sec((3*π)/4)=−√(,2)

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