Find the Exact Value sec(105)

Problem

sec(105)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

sec(105)=sec(60+45)

2. Use the reciprocal identity to rewrite the secant function in terms of the cosine function.

sec(105)=1/cos(60+45)

3. Apply the cosine sum formula, which is cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

cos(60+45)=cos(60)*cos(45)−sin(60)*sin(45)

4. Substitute the exact values for the trigonometric functions of 60 and 45

cos(60+45)=(1/2)*(√(,2)/2)−(√(,3)/2)*(√(,2)/2)

5. Simplify the expression for the cosine value.

cos(105)=√(,2)/4−√(,6)/4

cos(105)=(√(,2)−√(,6))/4

6. Find the reciprocal to determine the value of the secant function.

sec(105)=4/(√(,2)−√(,6))

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

sec(105)=(4*(√(,2)+√(,6)))/((√(,2)−√(,6))*(√(,2)+√(,6)))

sec(105)=(4*(√(,2)+√(,6)))/(2−6)

sec(105)=(4*(√(,2)+√(,6)))/(−4)

8. Simplify the final fraction.

sec(105)=−(√(,2)+√(,6))

Final Answer

sec(105)=−√(,2)−√(,6)

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