Find the Exact Value tan(42)

Problem

tan(42^∘)

Solution

1. Identify the expression as the tangent of an angle measured in degrees.

2. Determine if the angle 42^∘ is a standard reference angle (like 30^∘ 45^∘ or 60^∘. Since it is not, an exact value in terms of simple radicals requires using trigonometric identities or half-angle formulas.

3. Relate 42^∘ to known angles. Note that 42^∘=72^∘−30^∘ or 42^∘=60^∘−18^∘

4. Use the exact value for tan(18^∘) which is √(,1−2/√(,5))

5. Apply the tangent subtraction formula tan(A−B)=(tan(A)−tan(B))/(1+tan(A)*tan(B)) with A=60^∘ and B=18^∘

6. Substitute tan(60^∘)=√(,3) and tan(18^∘)=√(,5−2√(,5))/√(,5)

7. Simplify the resulting expression to reach the exact radical form.

Final Answer

tan(42^∘)=(√(,3)−√(,1−2/√(,5)))/(1+√(,3)√(,1−2/√(,5)))

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