Find the Exact Value csc(15)

Problem

csc(15^∘)

Solution

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

csc(15^∘)=1/sin(15^∘)

sin(15^∘)=sin(45^∘−30^∘)

2. Apply the sine difference formula sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B) to find the denominator.

sin(45^∘−30^∘)=sin(45^∘)*cos(30^∘)−cos(45^∘)*sin(30^∘)

3. Substitute the known values for the trigonometric functions of 45^∘ and 30^∘

sin(15^∘)=√(,2)/2⋅√(,3)/2−√(,2)/2⋅1/2

sin(15^∘)=(√(,6)−√(,2))/4

4. Invert the result to find the cosecant value.

csc(15^∘)=4/(√(,6)−√(,2))

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

csc(15^∘)=(4*(√(,6)+√(,2)))/((√(,6)−√(,2))*(√(,6)+√(,2)))

csc(15^∘)=(4*(√(,6)+√(,2)))/(6−2)

csc(15^∘)=(4*(√(,6)+√(,2)))/4

6. Simplify the expression by canceling the common factor.

csc(15^∘)=√(,6)+√(,2)

Final Answer

csc(15^∘)=√(,6)+√(,2)

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