1. **Problem Statement:** Construct the Fourier expansion for the function $y$ in terms of $x$ up to the first harmonic, given the data points: | $x^\circ$ | 0 | 60 | 120 | 180 | 240 | 300 | 360 | |---|---|---|---|---|---|---|---| | $y$ | 10.5 | 26.4 | 27 | 12.5 | -19.2 | -15.8 | 10.5 | 2. **Fourier Series Formula:** For a function with period $T=360^\circ$, the Fourier series up to the first harmonic is: $$y(x) = a_0 + a_1 \cos\left(\frac{2\pi x}{T}\right) + b_1 \sin\left(\frac{2\pi x}{T}\right)$$ where $$a_0 = \frac{1}{N} \sum_{k=0}^{N-1} y_k$$ $$a_1 = \frac{2}{N} \sum_{k=0}^{N-1} y_k \cos\left(\frac{2\pi k}{N}\right)$$ $$b_1 = \frac{2}{N} \sum_{k=0}^{N-1} y_k \sin\left(\frac{2\pi k}{N}\right)$$ with $N=6$ intervals (since $360^\circ$ divided by $60^\circ$ steps). 3. **Calculate $a_0$:** $$a_0 = \frac{1}{7} (10.5 + 26.4 + 27 + 12.5 - 19.2 - 15.8 + 10.5) = \frac{51.9}{7} = 7.4143$$ 4. **Calculate $a_1$ and $b_1$:** Convert $x$ to multiples of $60^\circ$, so $k=0$ to $6$. Calculate each term: - $a_1 = \frac{2}{7} \sum_{k=0}^6 y_k \cos\left(\frac{2\pi k}{7}\right)$ - $b_1 = \frac{2}{7} \sum_{k=0}^6 y_k \sin\left(\frac{2\pi k}{7}\right)$ Using values: $\cos\left(\frac{2\pi k}{7}\right)$ and $\sin\left(\frac{2\pi k}{7}\right)$ for $k=0$ to $6$ are: | k | cos | sin | |---|-----|-----| | 0 | 1 | 0 | | 1 | 0.6235 | 0.7818 | | 2 | -0.2225 | 0.9749 | | 3 | -0.9009 | 0.4339 | | 4 | -0.9009 | -0.4339 | | 5 | -0.2225 | -0.9749 | | 6 | 0.6235 | -0.7818 | Calculate sums: $$\sum y_k \cos\left(\frac{2\pi k}{7}\right) = 10.5(1) + 26.4(0.6235) + 27(-0.2225) + 12.5(-0.9009) + (-19.2)(-0.9009) + (-15.8)(-0.2225) + 10.5(0.6235)$$ $$= 10.5 + 16.45 - 6.01 - 11.26 + 17.29 + 3.52 + 6.55 = 37.04$$ $$a_1 = \frac{2}{7} \times 37.04 = 10.58$$ Similarly for $b_1$: $$\sum y_k \sin\left(\frac{2\pi k}{7}\right) = 10.5(0) + 26.4(0.7818) + 27(0.9749) + 12.5(0.4339) + (-19.2)(-0.4339) + (-15.8)(-0.9749) + 10.5(-0.7818)$$ $$= 0 + 20.63 + 26.32 + 5.42 + 8.33 + 15.39 - 8.21 = 67.88$$ $$b_1 = \frac{2}{7} \times 67.88 = 19.39$$ 5. **Final Fourier expansion up to first harmonic:** $$\boxed{y(x) = 7.4143 + 10.58 \cos\left(\frac{2\pi x}{360}\right) + 19.39 \sin\left(\frac{2\pi x}{360}\right)}$$ This formula approximates the given data using the first harmonic of the Fourier series.