Given a right triangle where angles $A$ and $B$ are acute and $A$ is supplementary to $B$, which of the following relationships is false?

• A. $\sin A=\sin B$
• B. $\cos A=\sin B$
• C. $\tan A=\frac{1}{\tan B}$
• D. $\sin A=\cos B$

Answer: (A)

Verified step by step guidance

1. Recall that in a right triangle, the two acute angles A and B satisfy the equation \(A + B = 90^\circ\) because the sum of angles in a triangle is \(180^\circ\) and one angle is \(90^\circ\).
2. Understand that if \(A\) and \(B\) are complementary angles (i.e., \(A + B = 90^\circ\)), then the trigonometric functions of these angles have specific relationships: \(\sin A = \cos B\), \(\cos A = \sin B\), and \(\tan A = \frac{1}{\tan B}\).
3. Note that the problem states \(A\) is supplementary to \(B\), which means \(A + B = 180^\circ\). Since both \(A\) and \(B\) are acute angles in a right triangle, this cannot be true because two acute angles cannot sum to \(180^\circ\).
4. Evaluate each given relationship under the assumption that \(A\) and \(B\) are complementary (which is true for a right triangle):
5. - \(\sin A = \sin B\) (Is this true for complementary angles?)
6. - \(\cos A = \sin B\) (This matches the complementary angle identity.)
7. - \(\tan A = \frac{1}{\tan B}\) (This also matches the complementary angle identity.)
8. - \(\sin A = \cos B\) (This is a standard complementary angle identity.)
9. Identify which relationship does not hold true for complementary angles in a right triangle. Since \(\sin A = \sin B\) is generally false for complementary acute angles, this is the false relationship.