Find the Exact Value csc(45)

Problem

csc(45^∘)

Solution

1. Identify the definition of the cosecant function in terms of the sine function.

csc(θ)=1/sin(θ)

2. Substitute the given angle of 45^∘ into the identity.

csc(45^∘)=1/sin(45^∘)

3. Recall the exact value of sin(45^∘) from the unit circle or a special right triangle.

sin(45^∘)=√(,2)/2

4. Substitute the value of sin(45^∘) back into the cosecant expression.

csc(45^∘)=1/√(,2)/2

5. Simplify the fraction by multiplying by the reciprocal.

csc(45^∘)=2/√(,2)

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

csc(45^∘)=(2√(,2))/2

7. Simplify the resulting expression by canceling the common factor of 2

csc(45^∘)=√(,2)

Final Answer

csc(45^∘)=√(,2)

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