Find the Exact Value csc(390)

Problem

csc(390^∘)

Solution

1. Find the coterminal angle by subtracting 360^∘ from the given angle to find an equivalent angle within the first rotation.

390^∘−360^∘=30^∘

2. Apply the reciprocal identity for the cosecant function, which states that csc(θ)=1/sin(θ)

csc(390^∘)=csc(30^∘)

csc(30^∘)=1/sin(30^∘)

3. Substitute the known value for sin(30^∘) which is 1/2 based on the unit circle or a 30−60−90 reference triangle.

sin(30^∘)=1/2

4. Simplify the fraction by calculating the reciprocal of 1/2

1/(1/2)=2

Final Answer

csc(390^∘)=2

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