Find the Exact Value tan(22.5)

Problem

tan(22.5)

Solution

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

2. Apply the formula for the half-angle of 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. Evaluate the trigonometric functions using the known values cos(45)=√(,2)/2 and sin(45)=√(,2)/2

tan(22.5)=(1−√(,2)/2)/√(,2)/2

5. Simplify the expression by multiplying the numerator and the denominator by 2

tan(22.5)=(2−√(,2))/√(,2)

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

tan(22.5)=((2−√(,2))√(,2))/2

7. Distribute the √(,2) in the numerator.

tan(22.5)=(2√(,2)−2)/2

8. Divide each term by 2 to reach the final exact value.

tan(22.5)=√(,2)−1

Final Answer

tan(22.5)=√(,2)−1

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