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Which relationship in the triangle must be true?
Triangle A B C is shown. Angle A C B is a right angle. The length of side C B is a, the length of side A C is b, and the length of side A B is c.
sin(B) = sin(A)
sin(B) = cos(90 – B)
cos(B) = sin(180 – B)
cos(B) = cos(A)
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Answer: sin(B) = cos(90° − B).
Reason: C = 90° so A + B = 90° ⇒ A = 90° − B. Using the cofunction identity cos(90° − B) = sin(B) gives sin(B) = cos(90° − B). The other equalities would force sin(B)=cos(B) (or sin(B)=sin(B) only in special cases like B=45°), so they are not generally true.
Reason: C = 90° so A + B = 90° ⇒ A = 90° − B. Using the cofunction identity cos(90° − B) = sin(B) gives sin(B) = cos(90° − B). The other equalities would force sin(B)=cos(B) (or sin(B)=sin(B) only in special cases like B=45°), so they are not generally true.
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