Find the Exact Value sec(pi/4)

Problem

sec(π/4)

Solution

1. Identify the definition of the secant function in terms of the cosine function.

sec(θ)=1/cos(θ)

2. Substitute the given angle π/4 into the identity.

sec(π/4)=1/cos(π/4)

3. Recall the exact value of cos(π/4) from the unit circle or a special right triangle.

cos(π/4)=√(,2)/2

4. Substitute the value of the cosine function back into the expression.

sec(π/4)=1/√(,2)/2

5. Simplify the fraction by multiplying by the reciprocal.

sec(π/4)=2/√(,2)

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

sec(π/4)=(2√(,2))/2

7. Cancel the common factor of 2 to find the final exact value.

sec(π/4)=√(,2)

Final Answer

sec(π/4)=√(,2)

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