Find the Exact Value csc(20)

Problem

csc(20)

Solution

1. Identify the expression as the cosecant of an angle measured in degrees, csc(20)

2. Relate the cosecant function to the sine function using the reciprocal identity csc(θ)=1/sin(θ)

3. Recognize that 20 is not a standard reference angle (like 30 45 or 60 for which the sine value is expressed in simple radicals.

4. Apply the triple angle formula for sine, sin(3*θ)=3*sin(θ)−4*sin3(θ) where θ=20 and 3*θ=60

5. Substitute the known value sin(60)=√(,3)/2 into the equation √(,3)/2=3*sin(20)−4*sin3(20)

6. Determine that solving this cubic equation for sin(20) involves complex numbers or nested radicals (using Cardano's formula), which does not simplify to a standard "exact value" form typically expected in trigonometry without using the original trigonometric expression or its inverse.

7. Conclude that the exact value is most commonly represented in its simplest trigonometric form or as the reciprocal of the root of the cubic equation.

Final Answer

csc(20)=csc(20)

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