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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