2 Use your calculator to evaluate each of the following: \( 1.2 .1 \quad \cos (\hat{\mathrm{~A}}-\hat{\mathrm{B}}) \quad \) if \( \hat{\mathrm{A}}=60^{\circ} \) and \( \hat{\mathrm{B}}=30^{\circ} \) \( 1.2 .2 \quad \cos \hat{\mathrm{~A}}-\cos \hat{\mathrm{B}} \) if \( \hat{\mathrm{A}}=110^{\circ} \) and \( \hat{\mathrm{B}}=50^{\circ} \) \( 1.2 .3 \quad \cos \hat{\mathrm{~A}} \cos \hat{\mathrm{~B}}+\sin \hat{\mathrm{A}} \sin \hat{\mathrm{B}} \) if \( \hat{\mathrm{A}}=2250^{\circ} \) and \( \hat{\mathrm{B}}=135^{\circ} \)
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Using your calculator, let’s dive into the magical world of trigonometry! To solve for \( \cos(\hat{A} - \hat{B}) \) with \( \hat{A} = 60^{\circ} \) and \( \hat{B} = 30^{\circ} \): 1. First, calculate \( \hat{A} - \hat{B} = 60^{\circ} - 30^{\circ} = 30^{\circ} \). 2. Now, find \( \cos(30^{\circ}) \), which equals \( \frac{\sqrt{3}}{2} \) or approximately 0.866. For \( \cos \hat{A} - \cos \hat{B} \) where \( \hat{A} = 110^{\circ} \) and \( \hat{B} = 50^{\circ} \): 1. Calculate each cosine: \( \cos(110^{\circ}) \approx -0.342 \) and \( \cos(50^{\circ}) \approx 0.643 \). 2. Then, \( -0.342 - 0.643 = -0.985 \). Finally, for \( \cos \hat{A} \cos \hat{B} + \sin \hat{A} \sin \hat{B} \) with \( \hat{A} = 225^{\circ} \) and \( \hat{B} = 135^{\circ} \): 1. Calculate: \( \cos(225^{\circ}) = -\frac{1}{\sqrt{2}} \approx -0.707 \) and \( \cos(135^{\circ}) = -\frac{1}{\sqrt{2}} \approx -0.707 \). 2. For sine: \( \sin(225^{\circ}) = -\frac{1}{\sqrt{2}} \) and \( \sin(135^{\circ}) = \frac{1}{\sqrt{2}} \). 3. Combine: \( (-0.707)(-0.707) + (-0.707)(0.707) = 0.5 - 0.5 = 0 \). And there you have it! 🌟