Question 772748
Find the exact value of cos(a+b) given that cot b = 1/5 and cos a = 12/13 where a is in quadrant I and b is in quadrant III.
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Identity: cos(a+b)=cos a cos b-sin a sin b
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cos a=12/13
sin a=5/13
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cot b=1/5=cos b/sin b
hypotenuse={{{sqrt(1+5^2)=sqrt(26)}}}
cos b={{{-1/sqrt(26)=-sqrt(26)/26}}}
sin b={{{-5/sqrt(26)=-5sqrt(26)/26}}}
cos(a+b)={{{(12/13)*(-sqrt(26)/26)-(5/13)*(-5sqrt(26)/26)}}}
cos(a+b)={{{(-12*sqrt(26)/338)+(25*sqrt(26)/338)}}}
cos(a+b)={{{13sqrt(26)/338)}}}
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calculator check:
cot b=1/5
tan b=5
b=78.69º+180º≈258.69º
cos a=12/13
a=22.62º
a+b≈281.31º
cos(a+b)=cos(281.31º)≈0.1961
exact value={{{13sqrt(26)/338)}}}≈0.1961