Question

Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12

Accepted Answer

Identify the angle given: $$ \frac{5\pi}{12} . $$Notice that $$ \frac{5\pi}{12} $$ can be expressed as a sum or difference of angles whose tangent values are known. For example, $$ \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} $$ because $$ \frac{\pi}{3} = \frac{4\pi}{12} $$ and $$ \frac{\pi}{4} = \frac{3\pi}{12} $$, and their sum is $$ \frac{7\pi}{12} $$, so check carefully for the correct decomposition. Find two angles $$ A $$ and $$ B $$ such that $$ \frac{5\pi}{12} = A + B  or$$ $$ A - B $$, where $$ A $$ and $$ B $$ are standard angles with known tangent values. For example, $$ \frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6} $$ because $$ \frac{\pi}{4} = \frac{3\pi}{12} $$ and $$ \frac{\pi}{6} = \frac{2\pi}{12} $$, and their sum is $$ \frac{5\pi}{12} . $$Use the tangent sum formula: $$ 	an(A + B) = \frac{	an A + 	an B}{1 - 	an A 	an B} . $$Substitute $$ A = \frac{\pi}{4} $$ and $$ B = \frac{\pi}{6} $$ into this formula. Recall the exact values: $$ 	an \frac{\pi}{4} = 1 $$ and $$ 	an \frac{\pi}{6} = \frac{1}{\sqrt{3}} . $$Substitute these values into the formula to express $$ 	an \frac{5\pi}{12}  in $$terms of known numbers. Simplify the resulting expression by performing the addition and subtraction in the numerator and denominator to get the exact value of $$ 	an \frac{5\pi}{12} .$$

Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12

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Concetti chiave

Sum and Difference Formulas for Tangent

Exact Values of Tangent for Special Angles

Angle Decomposition