Find the Exact Value tan(12)

Problem

tan(12)

Solution

1. Identify the expression as the tangent of an angle in degrees, tan(12)

2. Apply the half-angle formula for tangent, which is tan(θ/2)=(1−cos(θ))/sin(θ) using θ=24

3. Recognize that 24 can be expressed as 60−36

4. Use the values for sin(36)=√(,10−2√(,5))/4 and cos(36)=(1+√(,5))/4

5. Apply the difference formulas for sine and cosine to find sin(24) and cos(24)

sin(24)=sin(60)*cos(36)−cos(60)*sin(36)

cos(24)=cos(60)*cos(36)+sin(60)*sin(36)

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

tan(12)=(1−(cos(60)*cos(36)+sin(60)*sin(36)))/(sin(60)*cos(36)−cos(60)*sin(36))

7. Simplify the expression using the exact values sin(60)=√(,3)/2 and cos(60)=1/2

tan(12)=√(,(1−cos(24))/(1+cos(24)))

8. Evaluate the radical form or the nested radical expression.

tan(12)=√(,1−2/√(,5))⋅√(,3)+√(,5)−√(,2*(5+√(,5)))

9. Simplify to the standard radical form.

tan(12)=√(,5+2√(,5))−√(,3)−1

Final Answer

tan(12)=√(,5+2√(,5))−√(,3)−1

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