Find the Exact Value sec(pi/5)

Problem

sec(π/5)

Solution

1. Identify the relationship between the secant function and the cosine function.

sec(π/5)=1/cos(π/5)

2. Recall the exact value for cos(π/5) which corresponds to cos(36)

cos(π/5)=(1+√(,5))/4

3. Substitute this value into the reciprocal expression.

sec(π/5)=1/(1+√(,5))/4

4. Simplify the fraction by multiplying by the reciprocal.

sec(π/5)=4/(1+√(,5))

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1−√(,5)

sec(π/5)=(4*(1−√(,5)))/((1+√(,5))*(1−√(,5)))

6. Expand the denominator using the difference of squares.

sec(π/5)=(4−4√(,5))/(1−5)

7. Divide the terms by the resulting denominator −4

sec(π/5)=(4−4√(,5))/(−4)

8. Simplify the final expression.

sec(π/5)=−1+√(,5)

Final Answer

sec(π/5)=√(,5)−1

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