Find the Exact Value tan(45-30)

Problem

tan(45−30)

Solution

1. Identify the difference formula for the tangent function, which is tan(A−B)=(tan(A)−tan(B))/(1+tan(A)*tan(B))

2. Substitute the values A=45 and B=30 into the formula.

tan(45−30)=(tan(45)−tan(30))/(1+tan(45)*tan(30))

3. Evaluate the trigonometric values for the known angles tan(45)=1 and tan(30)=√(,3)/3

tan(45−30)=(1−√(,3)/3)/(1+(1)*(√(,3)/3))

4. Simplify the numerator and denominator by finding a common denominator of 3

tan(45−30)=(3−√(,3))/3/(3+√(,3))/3

5. Cancel the common denominator of 3 from the fraction.

tan(45−30)=(3−√(,3))/(3+√(,3))

6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 3−√(,3)

tan(45−30)=((3−√(,3))*(3−√(,3)))/((3+√(,3))*(3−√(,3)))

7. Expand the numerator using the square of a binomial and the denominator using the difference of squares.

tan(45−30)=(9−6√(,3)+3)/(9−3)

8. Combine like terms and simplify the resulting fraction.

tan(45−30)=(12−6√(,3))/6

tan(45−30)=2−√(,3)

Final Answer

tan(45−30)=2−√(,3)

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