Question
b) \( \cos ^{2} 45^{\circ}+4 \sin 30^{\circ} \)
Ask by Lindsey Gonzalez. in South Africa
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The final answer is \( \frac{5}{2} \).
Solution
To solve the expression cos²(45°) + 4 sin(30°), follow these steps:
1. Evaluate cos(45°): cos(45°) = √2/2, so cos²(45°) = (√2/2)² = 2/4 = 1/2.
2. Evaluate sin(30°): sin(30°) = 1/2.
3. Substitute the values into the expression: (1/2) + 4*(1/2) = (1/2) + 2 = 5/2.
Thus, the final answer is 5/2.
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Simplify this solution The Deep Dive
To solve the expression \( \cos^{2} 45^{\circ} + 4 \sin 30^{\circ} \), start by recalling that \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \). Thus, \( \cos^{2} 45^{\circ} = \left(\frac{\sqrt{2}}{2}\right)^{2} = \frac{2}{4} = \frac{1}{2} \). Next, \( \sin 30^{\circ} = \frac{1}{2} \), so \( 4 \sin 30^{\circ} = 4 \cdot \frac{1}{2} = 2 \). Finally, add the two results: \[ \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2}. \] Thus, the final answer is \( \frac{5}{2} \).
