Subjects trigonometry

Cosine Sine Acute 61Bd4C

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1. **Problem statement:** Determine if the statement "If \(\angle M\) and \(\angle N\) are acute angles in a right triangle, then \(\cos(M) = \sin(N)\)" is always, sometimes, or never true. 2. **Recall the properties of a right triangle:** In a right triangle, the sum of the two acute angles is \(90^\circ\), so \(M + N = 90^\circ\). 3. **Use the complementary angle identity:** For any angle \(\theta\), \(\sin(90^\circ - \theta) = \cos(\theta)\). 4. **Apply this to the problem:** Since \(N = 90^\circ - M\), then \(\sin(N) = \sin(90^\circ - M) = \cos(M)\). 5. **Conclusion:** The statement \(\cos(M) = \sin(N)\) is true whenever \(M\) and \(N\) are acute angles in a right triangle. **Final answer:** The statement is **always** true.