Find the Exact Value tan(112.5)

Problem

tan(112.5)

Solution

1. Identify the angle as a half-angle of a known special angle.

θ=112.5

2*θ=225

2. Apply the half-angle formula for tangent, using the version tan(α/2)=(1−cos(α))/sin(α)

tan(112.5)=(1−cos(225))/sin(225)

3. Determine the values of sine and cosine for 225 which is in the third quadrant.

cos(225)=−√(,2)/2

sin(225)=−√(,2)/2

4. Substitute these values into the half-angle formula.

tan(112.5)=(1−(−√(,2)/2))/(−√(,2)/2)

5. Simplify the numerator by finding a common denominator.

tan(112.5)=(2+√(,2))/2/(−√(,2)/2)

6. Divide the fractions by multiplying by the reciprocal of the denominator.

tan(112.5)=(2+√(,2))/2⋅(−2/√(,2))

tan(112.5)=−(2+√(,2))/√(,2)

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

tan(112.5)=−((2+√(,2))√(,2))/2

tan(112.5)=−(2√(,2)+2)/2

8. Simplify the final expression by dividing each term in the numerator by 2.

tan(112.5)=−(√(,2)+1)

Final Answer

tan(112.5)=−1−√(,2)

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