Find the Exact Value csc(75)

Problem

csc(75)

Solution

1. Identify the trigonometric identity for the cosecant function, which is the reciprocal of the sine function.

csc(75)=1/sin(75)

2. Rewrite the angle 75 as a sum of two special angles whose trigonometric values are known.

75=45+30

3. Apply the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) to find the value of sin(75)

sin(75)=sin(45+30)

sin(75)=sin(45)*cos(30)+cos(45)*sin(30)

4. Substitute the known values sin(45)=√(,2)/2 cos(30)=√(,3)/2 cos(45)=√(,2)/2 and sin(30)=1/2

sin(75)=√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2

sin(75)=√(,6)/4+√(,2)/4

sin(75)=(√(,6)+√(,2))/4

5. Calculate the reciprocal to find csc(75)

csc(75)=4/(√(,6)+√(,2))

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

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

csc(75)=(4*(√(,6)−√(,2)))/(6−2)

csc(75)=(4*(√(,6)−√(,2)))/4

csc(75)=√(,6)−√(,2)

Final Answer

csc(75)=√(,6)−√(,2)

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