Given: sine of cap A is equal to 5 over 13 , pi over 2 is less than cap A comma cap A is less than pi and tangent of cap b is equal to negative square root of 13 , pi over 2 is less than cap b comma cap b is less than pi What is tangent of open paren cap A minus cap b close paren ?
Answer
A
the fraction with numerator negative 5 plus 12 square root of 13 and denominator 12 plus 5 square root of 13
B
the fraction with numerator 12 plus 5 square root of 13 and denominator negative 5 plus 12 square root of 13
C
the fraction with numerator 5 plus 12 square root of 13 and denominator 12 minus 5 square root of 13
D
the fraction with numerator 12 minus 5 square root of 13 and denominator 5 plus 12 square root of 13
Explanation
Since A is in quadrant II and sin A = 5/13, its cosine is −12/13, so tan A = −5/12. Use tan(A − B) = (tan A − tan B)/(1 + tan A tan B), with tan B = −√13: this gives (−5/12 + √13)/(1 + 5√13/12). Multiplying the numerator and denominator by 12 gives (−5 + 12√13)/(12 + 5√13), which is choice A.
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