Question 759397
If sinA= -12/13, Q3, and cosB=4/5, Q1, calculate the exact value:
a) sec (A-B)
b) cot (A-B)
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sinA=-12/13
cosA=-5/13 (5-12-13 reference right triangle in Q3)
tanA=sin/cos=12/5
..
cosB=4/5
sinB=3/5 (3-4-5  reference right triangle in Q1)
tanB=sin/cos=3/4
..
Identity:{{{cos(A-B)=cosAcosB+sinAsinB=-5/13*4/5-12/13*3/5=-20/65-36/65=-56/65}}}
a) {{{sec (A-B)=1/cos(A-B)=-65/56}}}
..
Identity:{{{tan(A-B)=(tanA-tanB)/(1+tanAtanB)=(12/5-3/4)/(1+12/5*3/4)
=(48/20-15/20)/(1+36/20)=(48/20-15/20)/(56/20)=33/56}}}
b) {{{cot (A-B)=1/tan(A-B)=56/33}}}
computer check:
sinA=-12/13
A≈67.38+180≈247.38º
cosB=4/5
B≈36.87º
A-B≈247.38-36.87≈210.51º
..
cos(A-B)=cos(210.51º)≈-0.8615
exact value=-56/65≈-0.8615
..
tan(A-B)=tan(210.51º)≈0.5892
exact value=33/56≈0.5892