Evaluate csc(405 degrees )

Problem

csc(405)

Solution

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

405−360=45

2. Rewrite the expression using the coterminal angle, as trigonometric functions of coterminal angles have the same value.

csc(405)=csc(45)

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

csc(45)=1/sin(45)

4. Substitute the known value for sin(45) which is √(,2)/2

1/sin(45)=1/√(,2)/2

5. Simplify the fraction by taking the reciprocal of the denominator.

1/√(,2)/2=2/√(,2)

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

(2√(,2))/2=√(,2)

Final Answer

csc(405)=√(,2)

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