Answer to Question #167792 in Civil and Environmental Engineering for eyel

Question #167792

1.      Given that csc A = , A in QI, and sec B = , sin B < 0, find

a.      cos (A – B)

b.     tan (A – B)


1
Expert's answer
2021-03-04T06:02:57-0500
"\\csc A=\\dfrac{1}{\\sin A}=a, a\\geq 1"


"\\sin A=\\dfrac{1}{a}, "

"\\cos A=\\sqrt{1-\\sin^2 A}=\\sqrt{1-(\\dfrac{1}{a})^2}=\\dfrac{\\sqrt{a^2-1}}{a}"

"\\sec B=\\dfrac{1}{\\cos B}=b, |b|\\geq 1"

"\\cos B=\\dfrac{1}{b}"

"\\sin B=-\\sqrt{1-\\cos^2 B}=-\\sqrt{1-(\\dfrac{1}{b})^2}=-\\dfrac{\\sqrt{b^2-1}}{|b|}"

a.


"\\cos(A-B)=\\cos A\\cos B+\\sin A\\sin B"

"=\\dfrac{\\sqrt{a^2-1}}{a}(\\dfrac{1}{b})+\\dfrac{1}{a}(-\\dfrac{\\sqrt{b^2-1}}{|b|})"

"b\\geq1, a\\geq1:\\cos(A-B)=\\dfrac{\\sqrt{a^2-1}-\\sqrt{b^2-1}}{ab}"


"b\\leq-1, a\\geq1:\\cos(A-B)=\\dfrac{\\sqrt{a^2-1}+\\sqrt{b^2-1}}{ab}"

b.


"\\tan(A-B)=\\dfrac{\\sin(A-B)}{\\cos(a-B)}"

"\\sin(A-B)=\\sin A\\cos B-\\cos A\\sin B"

"=\\dfrac{1}{a}(\\dfrac{1}{b})-\\dfrac{\\sqrt{a^2-1}}{a}(-\\dfrac{\\sqrt{b^2-1}}{|b|})"

"b\\geq1, a\\geq1:\\cos(A-B)=\\dfrac{\\sqrt{a^2-1}-\\sqrt{b^2-1}}{ab}"


"\\sin(A-B)=\\dfrac{1+\\sqrt{(a^2-1)(b^2-1)}}{ab}"

"\\tan(A-B)=\\dfrac{1+\\sqrt{(a^2-1)(b^2-1)}}{\\sqrt{a^2-1}-\\sqrt{b^2-1}}"


"b\\leq-1, a\\geq1:\\cos(A-B)=\\dfrac{\\sqrt{a^2-1}+\\sqrt{b^2-1}}{ab}"


"\\sin(A-B)=\\dfrac{1-\\sqrt{(a^2-1)(b^2-1)}}{ab}"

"\\tan(A-B)=\\dfrac{1-\\sqrt{(a^2-1)(b^2-1)}}{\\sqrt{a^2-1}+\\sqrt{b^2-1}}"


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