Find the Exact Value sec(75)

Problem

sec(75)

Solution

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

75=45+30

2. Apply the reciprocal identity for secant.

sec(75)=1/cos(75)

3. Use the cosine sum formula cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) to find the denominator.

cos(45+30)=cos(45)*cos(30)−sin(45)*sin(30)

4. Substitute the known values for the trigonometric functions.

cos(75)=√(,2)/2⋅√(,3)/2−√(,2)/2⋅1/2

5. Simplify the expression for the cosine value.

cos(75)=(√(,6)−√(,2))/4

6. Invert the result to find the secant value.

sec(75)=4/(√(,6)−√(,2))

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

sec(75)=(4*(√(,6)+√(,2)))/((√(,6)−√(,2))*(√(,6)+√(,2)))

8. Simplify the denominator using the difference of squares.

sec(75)=(4*(√(,6)+√(,2)))/(6−2)

9. Cancel the common factor of 4.

sec(75)=(4*(√(,6)+√(,2)))/4

Final Answer

sec(75)=√(,6)+√(,2)

Want more problems? Check here!