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Calculating pi with sine of half angles

Calculating pi with sine of half angles
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2020-10-23 04:30
AngleTrigonometrySequenceLimitGeometric Construction
① This example demonstrates how to approximate π by repeatedly halving an angle and computing its sine, showing an iterative trigonometric approach to a fundamental constant. ② Core mathematical concepts include the half-angle formula, sine function, limits, and the geometric idea of approximating a circle by polygons, along with angle-radian conversion. ③ Construction highlights: using GeoGebra commands such as Rotate, Segment, Polygon, OrthogonalLine, Intersect, and Angle, the applet builds a series of rotated half-angle triangles, measures angle α and side lengths, and uses the sine values in an iterative product or recurrence to estimate π. ④ Suitable for high school or early college students to explore trigonometric identities, limits, and numerical approximation methods, and for use in math labs or inquiry-based learning.

GeoGebra Commands & Parameters

  1. Rotate(A, α, B) → A'
  2. Rotate(A, (-α), B) → A'_1
  3. Segment(A', A'_1) → f
  4. Polygon(A'_1, A', (4 * 2^(a - 1))) → poly1, g, q, r_5, s_5, t_5, a_5, b_5, c_5, d_5, e_5, f_6, g_6, r, h_6, i_6, j_6, k_6, l_6, m_6, n_6, p_6, q_6, r_6, s, s_6, t_6, a_6, b_6, c_6, d_6, e_6, f_7, g_7, h_7, t, i_7, j_7, k_7, l_7, m_7, n_7, p_7, q_7, r_7, s_7, b, t_7, a_7, b_7, c_7, d_7, e_7, f_8, g_8, h_8, i_8, d, j_8, k_8, l_8, m_8, n_8, p_8, q_8, r_8, s_8, t_8, e, a_8, b_8, c_8, d_8, e_8, f_9, g_9, h_9, i_9, j_9, f_1, k_9, l_9, m_9, n_9, p_9, q_9, r_9, s_9, t_9, a_9, g_1, b_9, c_9, d_9, e_9, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, h_1, l_{10}, m_{10}, n_{10}, p_{10}, q_{10}, r_{10}, s_{10}, t_{10}, a_{10}, b_{10}, h, i_1, c_{10}, d_{10}, e_{10}, f_{11}, g_{11}, h_{11}, i_{11}, j_{11}, k_{11}, l_{11}, j_1, m_{11}, n_{11}, p_{11}, q_{11}, r_{11}, s_{11}, t_{11}, a_{11}, b_{11}, c_{11}, k_1, d_{11}, e_{11}, f_{12}, g_{12}, h_{12}, i_{12}, j_{12}, k_{12}, l_{12}, m_{12}, l_1, n_{12}, p_{12}, q_{12}, r_{12}, s_{12}, t_{12}, a_{12}, b_{12}, c_{12}, d_{12}, m_1, e_{12}, f_{13}, g_{13}, h_{13}, i_{13}, j_{13}, k_{13}, l_{13}, m_{13}, n_{13}, n_1, p_{13}, q_{13}, r_{13}, s_{13}, t_{13}, a_{13}, b_{13}, C, D, E, p_1, F, G, H, I, J, K, L, M, N, O, q_1, P, Q, R, S, T, U, V, W, Z, A_1, r_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, K_1, s_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1, U_1, i, t_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2, E_2, F_2, G_2, a_1, H_2, I_2, J_2, K_2, L_2, M_2, N_2, O_2, P_2, T_2, b_1, U_2, V_2, W_2, Z_2, A_3, B_3, C_3, D_3, E_3, F_3, c_1, G_3, H_3, I_3, J_3, K_3, L_3, M_3, N_3, O_3, P_3, d_1, Q_3, R_3, S_3, T_3, U_3, V_3, W_3, Z_3, A_4, B_4, e_1, C_4, D_4, E_4, F_4, G_4, H_4, I_4, J_4, K_4, L_4, f_2, M_4, N_4, O_4, P_4, Q_4, R_4, S_4, T_4, U_4, V_4, g_2, W_4, Z_4, A_5, B_5, C_5, D_5, E_5, F_5, G_5, H_5, h_2, I_5, J_5, K_5, L_5, M_5, N_5, O_5, P_5, Q_5, R_5, i_2, S_5, T_5, U_5, V_5, W_5, Z_5, A_6, B_6, C_6, D_6, j, j_2, E_6, F_6, G_6, H_6, I_6, J_6, K_6, L_6, M_6, N_6, k_2, O_6, P_6, Q_6, R_6, S_6, T_6, U_6, V_6, W_6, Z_6, l_2, A_7, B_7, C_7, D_7, E_7, F_7, G_7, H_7, I_7, J_7, m_2, K_7, L_7, M_7, N_7, O_7, P_7, Q_7, R_7, S_7, T_7, n_2, U_7, V_7, W_7, Z_7, A_8, B_8, C_8, D_8, E_8, F_8, p_2, G_8, H_8, I_8, J_8, K_8, L_8, M_8, N_8, O_8, P_8, q_2, Q_8, R_8, S_8, T_8, U_8, V_8, W_8, Z_8, A_9, B_9, r_2, C_9, D_9, E_9, F_9, G_9, H_9, I_9, J_9, K_9, L_9, s_2, M_9, N_9, O_9, P_9, Q_9, R_9, S_9, T_9, U_9, V_9, t_2, W_9, Z_9, A_{10}, B_{10}, C_{10}, D_{10}, E_{10}, F_{10}, G_{10}, H_{10}, k, a_2, I_{10}, J_{10}, K_{10}, L_{10}, M_{10}, N_{10}, O_{10}, P_{10}, Q_{10}, R_{10}, b_2, S_{10}, c_2, d_2, e_2, f_3, g_3, h_3, i_3, j_3, l, k_3, l_3, m_3, n_3, p_3, a_3, b_3, c_3, d_3, e_3, m, f_4, g_4, h_4, i_4, j_4, k_4, l_4, m_4, n_4, p_4, n, q_4, r_4, s_4, t_4, a_4, b_4, c_4, d_4, e_4, f_5, p, g_5, h_5, i_5, j_5, k_5, l_5, m_5, n_5, p_5, q_5
  5. Segment(B, A'_1) → q_3
  6. Segment(B, A') → r_3
  7. OrthogonalLine(B, f) → s_3
  8. Intersect(f, g) → Q_2
  9. Intersect(f, g) → R_2
  10. Intersect(f, s_3) → S_2
  11. Segment(B, S_2) → t_3
  12. Angle(B, S_2, A'_1) → β

AI Drawing Prompt Reference

Draw a sequence of triangles with angles repeatedly halved, starting with angle α. First, rotate point B around point A by -α to get A', then connect A' and B to form line segment f. Construct a regular polygon centered at B with f as one of its sides. Continuously rotate and construct new triangles to form a series of half-angle triangles. Measure the angles and side lengths of each triangle, and use the sine values in an iterative process to approximate π.
The prompt above was compiled from the original drawing commands and is for reference only.