Simplify 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 (reference angle 45.

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 and the fraction.

tan(112.5)=(1+√(,2)/2)/(−√(,2)/2)

tan(112.5)=(2+√(,2))/2/(−√(,2)/2)

tan(112.5)=(2+√(,2))/(−√(,2))

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

tan(112.5)=((2+√(,2))√(,2))/(−√(,2)⋅√(,2))

tan(112.5)=(2√(,2)+2)/(−2)

tan(112.5)=−√(,2)−1

Final Answer

tan(112.5)=−1−√(,2)

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