Is this statement always, sometimes, or never true?If m∠D is greater than 35°, then cos(D)=sin(D).alwayssometimesneverSubmit

1. Break Down the Problem

We need to determine the validity of the statement: "If $m∠D > 35^\circ$, then $\cos(D) = \sin(D)$." This involves understanding the relationship between cosine and sine functions.

2. Relevant Concepts

The equation $\cos(D) = \sin(D)$ occurs when the angle $D$ is equal to $45^\circ$ (or $\frac{\pi}{4}$ radians). This is derived from the identity: $\cos(D) = \sin(D) \Rightarrow \tan(D) = 1 \Rightarrow D = 45^\circ + k \cdot 180^\circ \text{ (for any integer } k\text{)}$ Thus, for angles greater than $35^\circ$, we need to explore when $\cos(D)$ and $\sin(D)$ can be equal.

3. Analysis and Detail

1. Evaluate the angles:

   • At $D = 45^\circ$, $\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$
   • Between $35^\circ$ and $45^\circ$, $\cos(D)$ is greater than $\sin(D)$.
   • At $D = 45^\circ$, they are equal.
   • For angles greater than $45^\circ$ (up to $90^\circ$), $\cos(D) < \sin(D)$.

2. Conclusion from evaluations: Hence, the statement "If $m∠D > 35^\circ$, then $\cos(D) = \sin(D)$" is not consistently true since $D$ can be any angle greater than $35^\circ$ and might not equal $45^\circ$.

4. Verify and Summarize

By examining the angle conditions:

• It is true for $D = 45^\circ$, but false for values outside this specific angle within the range we considered.
• Therefore, the statement does not hold true for all angles greater than $35^\circ$.

Final Answer

The statement is sometimes true. It is true specifically when $m∠D = 45^\circ$ but not for all angles greater than $35^\circ$.