Find the Value Using the Unit Circle csc(45 degrees )

Problem

csc(45^∘)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(θ)=1/sin(θ)

2. Determine the sine of 45^∘ using the unit circle or a special right triangle (45^∘−45^∘−90^∘.

sin(45^∘)=√(,2)/2

3. Substitute the value of sin(45^∘) into the reciprocal identity.

csc(45^∘)=1/√(,2)/2

4. Simplify the expression by multiplying by the reciprocal of the denominator.

csc(45^∘)=2/√(,2)

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

csc(45^∘)=(2√(,2))/2

6. Reduce the fraction to find the final value.

csc(45^∘)=√(,2)

Final Answer

csc(45^∘)=√(,2)

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