Evaluate csc(420 degrees )

Problem

csc(420)

Solution

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

420−360=60

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

csc(60)=1/sin(60)

3. Determine the sine value for the reference angle 60 using the unit circle or special right triangles.

sin(60)=√(,3)/2

4. Substitute and simplify the expression by taking the reciprocal of the sine value.

csc(60)=1/√(,3)/2

csc(60)=2/√(,3)

5. Rationalize the denominator by multiplying the numerator and the denominator by √(,3)

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

csc(420)=(2√(,3))/3

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