2.2 Evaluate inthout using a calculator: \( \sum_{A=38^{\circ}}^{52^{\circ}} \cos ^{2} A \)

To evaluate the sum \( \sum_{A=38^{\circ}}^{52^{\circ}} \cos ^{2} A \), we will first express \( \cos^2 A \) in a more manageable form using the double angle identity: \[ \cos^2 A = \frac{1 + \cos(2A)}{2} \] Now, we can rewrite the sum: \[ \sum_{A=38^{\circ}}^{52^{\circ}} \cos^2 A = \sum_{A=38^{\circ}}^{52^{\circ}} \frac{1 + \cos(2A)}{2} \] This can be separated into two sums: \[ \sum_{A=38^{\circ}}^{52^{\circ}} \cos^2 A = \frac{1}{2} \sum_{A=38^{\circ}}^{52^{\circ}} 1 + \frac{1}{2} \sum_{A=38^{\circ}}^{52^{\circ}} \cos(2A) \] Now, let's evaluate each part: 1. **First Sum**: \( \sum_{A=38^{\circ}}^{52^{\circ}} 1 \) This sum counts the number of terms from \( 38^{\circ} \) to \( 52^{\circ} \). The terms are \( 38^{\circ}, 39^{\circ}, 40^{\circ}, 41^{\circ}, 42^{\circ}, 43^{\circ}, 44^{\circ}, 45^{\circ}, 46^{\circ}, 47^{\circ}, 48^{\circ}, 49^{\circ}, 50^{\circ}, 51^{\circ}, 52^{\circ} \), which gives us a total of \( 52 - 38 + 1 = 15 \) terms. 2. **Second Sum**: \( \sum_{A=38^{\circ}}^{52^{\circ}} \cos(2A) \) We will calculate this sum by evaluating \( \cos(2A) \) for each angle from \( 38^{\circ} \) to \( 52^{\circ} \): - \( 2 \times 38^{\circ} = 76^{\circ} \) - \( 2 \times 39^{\circ} = 78^{\circ} \) - \( 2 \times 40^{\circ} = 80^{\circ} \) - \( 2 \times 41^{\circ} = 82^{\circ} \) - \( 2 \times 42^{\circ} = 84^{\circ} \) - \( 2 \times 43^{\circ} = 86^{\circ} \) - \( 2 \times 44^{\circ} = 88^{\circ} \) - \( 2 \times 45^{\circ} = 90^{\circ} \) - \( 2 \times 46^{\circ} = 92^{\circ} \) - \( 2 \times 47^{\circ} = 94^{\circ} \) - \( 2 \times 48^{\circ} = 96^{\circ} \) - \( 2 \times 49^{\circ} = 98^{\circ} \) - \( 2 \times 50^{\circ} = 100^{\circ} \) - \( 2 \times 51^{\circ} = 102^{\circ} \) - \( 2 \times 52^{\circ} = 104^{\circ} \) Now we can calculate \( \sum_{A=38^{\circ}}^{52^{\circ}} \cos(2A) \): \[ \sum_{A=38^{\circ}}^{52^{\circ}} \cos(2A) = \cos(76^{\circ}) + \cos(78^{\circ}) + \cos(80^{\circ}) + \cos(82^{\circ}) + \cos(84^{\circ}) + \cos(86^{\circ}) + \cos(88^{\circ}) + \cos(90^{\circ}) + \cos(92^{\circ}) + \cos(94^{\circ}) + \cos(96^{\circ}) + \cos(98^{\circ}) + \cos(100^{\circ}) + \cos(102^{\circ}) + \cos(104^{\circ}) \] Using the property \( \cos(180^{\circ} - x) = -\cos(x) \), we can pair terms: - \( \cos(76^{\circ}) + \cos(104^{\circ}) = 0 \) - \( \cos(78^{\circ}) + \cos(102^{\circ}) = 0 \) - \( \cos(80^{\circ}) + \cos(100^{\circ}) = 0 \) - \( \cos(82^{\circ}) + \cos(98^{\circ}) = 0 \) - \( \cos(84^{\circ}) + \cos(96^{\circ}) = 0 \) - \( \cos(86^{\circ}) + \cos(94^{\circ}) = 0 \) - \( \cos(88^{\circ}) + \cos(92^{\circ}) = 0 \) - \( \cos(90^{\circ}) = 0 \) Thus, the sum \( \sum_{A=38^{\circ}}^{52^{\circ}} \cos(2A) = 0 \). Now we can combine our results: \[ \sum_{A=38^{\circ}}^{52^{\circ}} \cos^2 A = \frac{1}{2} \cdot 15 + \frac{1}{2} \cdot 0 = \frac{15}{2} \] Therefore, the final result is: \[ \sum_{A=38^{\circ}}^{52^{\circ}} \cos^2 A = 7.5 \]