Expand Using Sum/Difference Formulas sec((-11pi)/12)

Problem

sec(−(11*π)/12)

Solution

1. Apply the even-odd identity for the secant function, which states sec(−θ)=sec(θ)

sec(−(11*π)/12)=sec((11*π)/12)

2. Rewrite the angle as a sum or difference of common angles from the unit circle.

(11*π)/12=(3*π)/12+(8*π)/12

(11*π)/12=π/4+(2*π)/3

3. Use the reciprocal identity to express secant in terms of cosine.

sec(π/4+(2*π)/3)=1/cos(π/4+(2*π)/3)

4. Apply the cosine sum formula cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

cos(π/4+(2*π)/3)=cos(π/4)*cos((2*π)/3)−sin(π/4)*sin((2*π)/3)

5. Substitute the exact values for the trigonometric functions.

cos(π/4+(2*π)/3)=(√(,2)/2)*(−1/2)−(√(,2)/2)*(√(,3)/2)

cos(π/4+(2*π)/3)=−√(,2)/4−√(,6)/4

cos(π/4+(2*π)/3)=(−√(,2)−√(,6))/4

6. Find the reciprocal to determine the secant value.

sec((11*π)/12)=4/(−√(,2)−√(,6))

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

sec((11*π)/12)=(4*(−√(,2)+√(,6)))/((−√(,2)−√(,6))*(−√(,2)+√(,6)))

sec((11*π)/12)=(4*(√(,6)−√(,2)))/(2−6)

sec((11*π)/12)=(4*(√(,6)−√(,2)))/(−4)

sec((11*π)/12)=−(√(,6)−√(,2))

sec((11*π)/12)=√(,2)−√(,6)

Final Answer

sec(−(11*π)/12)=√(,2)−√(,6)

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