$$ 2 \cos \hat{A}-\cos \hat{B} $$ if $$ \hat{A}=110^{\circ} $$ and $$ \hat{B}=50^{\circ} $$

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1. Substitute $$\hat{A}=110^\circ$$ and $$\hat{B}=50^\circ$$ into the expression:
   $$
   2\cos(110^\circ)-\cos(50^\circ)
   $$

2. Recognize that $$\cos(110^\circ)$$ can be expressed using the identity $$\cos(180^\circ-\theta)=-\cos(\theta)$$. Since $$110^\circ=180^\circ-70^\circ$$, we have:
   $$
   \cos(110^\circ)=-\cos(70^\circ)
   $$

3. Replace $$\cos(110^\circ)$$ in the expression:
   $$
   2(-\cos(70^\circ))-\cos(50^\circ) = -2\cos(70^\circ)-\cos(50^\circ)
   $$

4. Compute the approximate values:
   $$
   \cos(70^\circ)\approx0.3420 \quad \text{and} \quad \cos(50^\circ)\approx0.6428
   $$

5. Substitute these approximations into the expression:
   $$
   -2(0.3420)-0.6428 = -0.6840-0.6428 \approx -1.3268
   $$

Thus, the value of the expression is approximately $$-1.327$$.
The value of $$ 2 \cos 110^\circ - \cos 50^\circ $$ is approximately $$-1.327$$.

\( 2 \cos \hat{A}-\cos \hat{B} \) if \( \hat{A}=110^{\circ} \) and \( \hat{B}=50^{\circ} \)

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