Find the Exact Value tan(63)

Problem

tan(63)

Solution

1. Identify the angle as 63 Since this is not a standard reference angle, we can express it as the sum of two angles: 45+18

2. Apply the sum formula for tangent, which is tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

3. Substitute A=45 and B=18 into the formula.

tan(63)=tan(45+18)

tan(63)=(tan(45)+tan(18))/(1−tan(45)*tan(18))

4. Evaluate the known value tan(45)=1

tan(63)=(1+tan(18))/(1−tan(18))

5. Determine the value of tan(18) Using the properties of a golden triangle, tan(18)=√(,5−2√(,5))/√(,5) This can also be written as tan(18)=√(,1−2/√(,5))

6. Substitute the value of tan(18) back into the expression.

tan(63)=(1+√(,1−2/√(,5)))/(1−√(,1−2/√(,5)))

7. Simplify the expression using the radical form tan(18)=√(,25−10√(,5))/5

tan(63)=(5+√(,25−10√(,5)))/(5−√(,25−10√(,5)))

Final Answer

tan(63)=(5+√(,25−10√(,5)))/(5−√(,25−10√(,5)))

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