Find the Value Using the Unit Circle tan(22.5)

Problem

tan(22.5^∘)

Solution

1. Identify the angle 22.5^∘ as a half-angle of a known unit circle value, since 22.5^∘=45^∘/2

2. Apply the half-angle formula for tangent, which is tan(θ/2)=(1−cos(θ))/sin(θ)

3. Substitute θ=45^∘ into the formula.

tan(22.5^∘)=(1−cos(45^∘))/sin(45^∘)

4. Use unit circle values for 45^∘ where cos(45^∘)=√(,2)/2 and sin(45^∘)=√(,2)/2

tan(22.5^∘)=(1−√(,2)/2)/√(,2)/2

5. Simplify the complex fraction by multiplying the numerator and denominator by 2

tan(22.5^∘)=(2−√(,2))/√(,2)

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

tan(22.5^∘)=((2−√(,2))√(,2))/2

tan(22.5^∘)=(2√(,2)−2)/2

tan(22.5^∘)=√(,2)−1

Final Answer

tan(22.5^∘)=√(,2)−1

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