Find the Exact Value tan(300+45)

Problem

tan(300+45)

Solution

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

2. Substitute the values A=300 and B=45 into the formula.

tan(300+45)=(tan(300)+tan(45))/(1−tan(300)*tan(45))

3. Determine the exact values of the trigonometric components. Since 300 is in the fourth quadrant with a reference angle of 60 tan(300)=−√(,3) For 45 tan(45)=1

tan(300)=−√(,3)

tan(45)=1

4. Plug these values back into the expression.

tan(300+45)=(−√(,3)+1)/(1−(−√(,3))*(1))

tan(300+45)=(1−√(,3))/(1+√(,3))

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1−√(,3)

tan(300+45)=((1−√(,3))*(1−√(,3)))/((1+√(,3))*(1−√(,3)))

tan(300+45)=(1−2√(,3)+3)/(1−3)

tan(300+45)=(4−2√(,3))/(−2)

6. Simplify the fraction by dividing each term in the numerator by −2

tan(300+45)=−2+√(,3)

Final Answer

tan(300+45)=√(,3)−2

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