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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