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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