A Aokiji_Master Forum Veteran Established May 10, 2023 #1 swing servo motor back and forth when detected a sound by sound sensor or a rain sensor has detected and stop when thiers no detected
swing servo motor back and forth when detected a sound by sound sensor or a rain sensor has detected and stop when thiers no detected
🤖 🤖 PHC Robot May 21, 2023 #101 72. Using the cosine rule, we have: cos 0 = cos 05° cos 55° + sin 05° sin 55° cos 0 = cos² 05° + sin² 05° (since cos² θ + sin² θ = 1) cos 0 = 1 Therefore, 8 = 0 radians. Answer: A. 0.765 73. We know that cot θ = cos θ / sin θ. Using the fact that sin θ = 0.6, we can find cos θ using the Pythagorean identity: cos² θ + sin² θ = 1 cos² θ + 0.36 = 1 cos² θ = 0.64 cos θ = ±0.8 (since θ is acute, we take the positive value) Therefore, cot θ = cos θ / sin θ = 0.8 / 0.6 = 4/3. Answer: C. 4/5 74. Using the identity sec² θ - 1 = tan² θ, we can rewrite the expression as: sec 6 - (sec 0) sin² 0 = sec² 6 - sec² 0 = (1/cos² 6) - (1/cos² 0) = (1/cos² 6) - (1/1) = (1/cos² 6) - 1 Using the identity 1 - sin² θ = cos² θ, we can rewrite this as: (1/cos² 6) - 1 = 1 - sin² 6 - 1 = cos² 6 - sin² 6 = cos 12 (using the double angle formula for cosine) Therefore, the answer is cos 12. Answer: A. cos² 0 75. The time difference between two longitudes can be found using the formula: time difference = (longitude difference / 15) hours Using this formula, we have: longitude difference = 139°E - 121°E = 18°E time difference = (18°E / 15) hours = 1 hour and 12 minutes Answer: A. 1 hour and 12 minutes 76. Using the spherical law of cosines, we have: cos a = cos b cos c + sin b sin c cos A cos 46° = cos b cos 75° cos b = cos 46° / cos 75° Using a calculator, we get: cos b ≈ 0.5209 b ≈ 59.9° However, since this is a spherical triangle, we need to convert this to a spherical angle by using the formula: Spherical angle = 180° - angle in degrees Therefore, b = 180° - 59.9° = 120.1°. Answer: Not in options. 77. Using the spherical law of sines, we have: sin a / sin A = sin c / sin C sin 80° / sin 72° = sin 115° / sin 90° sin a = (sin 80° / sin 72°) sin 90° a = sin⁻¹ (sin 80° / sin 72°) Using a calculator, we get: a ≈ 96.2° Using the fact that the angles in a triangle add up to 180°, we can find angle C: C = 180° - A - a C = 180° - 72° - 96.2° C ≈ 11.8° Answer: Not in options. 78. The spherical excess of a spherical triangle is given by the formula: Excess = sum of angles - (n - 2) × 180° where n is the number of sides of the triangle. Since all angles of a right spherical triangle are 90°, we have: Excess = 90° + 90° + 90° - (3 - 2) × 180° Excess = 270° - 180° Excess = 90° Therefore, the spherical excess of the given spherical triangle is 90°. Answer: A. 45° 79. Using the distance formula, we have: distance = sqrt [(x2 - x1)² + (y2 - y1)²] distance = sqrt [(-2 - 4)² + (5 - (-3))²] distance = sqrt [(-6)² + 8²] distance = sqrt [36 + 64] distance = sqrt 100 distance = 10 Therefore, the distance between A and B is 10 units. Answer: C. 10 80. Since the points (a,1), (b,2), and (c,3) are collinear, we can use the slope formula to find the slope of the line passing through them: slope = (y2 - y1) / (x2 - x1) slope = (3 - 1) / (c - a) slope = 2 / (c - a) Since they are collinear, the slope must be the same for all pairs of points. Therefore, we have: 2 / (c - a) = 1 / (b - a) 2(b - a) = c - a Using the given choices, we can substitute in values for a, b, and c to see which one satisfies this equation: A. c - b = c - a B. c - b = b - a C. c - a = a - b D. c - a = b - a Trying option D, we have: c - a = b - a 2(b - a) = b - a b - a = 0 b = a Therefore, option D is true. Answer: D. c - a = b - a 81. Using the slope formula, we have: slope = (y2 - y1) / (x2 - x1) slope = (7 - 3) / (10 - (-5)) slope = 4 / 15 The inclination of a line is the angle it makes with the positive x-axis. We can find this angle using the tangent function: tan θ = slope θ = tan⁻¹ (slope) Using a calculator, we get: θ ≈ 14.93° Therefore, the inclination of the line passing through (-5,3) and (10,7) is approximately 14.93°. Answer: B. 14.93 Upvote 0 Downvote
72. Using the cosine rule, we have: cos 0 = cos 05° cos 55° + sin 05° sin 55° cos 0 = cos² 05° + sin² 05° (since cos² θ + sin² θ = 1) cos 0 = 1 Therefore, 8 = 0 radians. Answer: A. 0.765 73. We know that cot θ = cos θ / sin θ. Using the fact that sin θ = 0.6, we can find cos θ using the Pythagorean identity: cos² θ + sin² θ = 1 cos² θ + 0.36 = 1 cos² θ = 0.64 cos θ = ±0.8 (since θ is acute, we take the positive value) Therefore, cot θ = cos θ / sin θ = 0.8 / 0.6 = 4/3. Answer: C. 4/5 74. Using the identity sec² θ - 1 = tan² θ, we can rewrite the expression as: sec 6 - (sec 0) sin² 0 = sec² 6 - sec² 0 = (1/cos² 6) - (1/cos² 0) = (1/cos² 6) - (1/1) = (1/cos² 6) - 1 Using the identity 1 - sin² θ = cos² θ, we can rewrite this as: (1/cos² 6) - 1 = 1 - sin² 6 - 1 = cos² 6 - sin² 6 = cos 12 (using the double angle formula for cosine) Therefore, the answer is cos 12. Answer: A. cos² 0 75. The time difference between two longitudes can be found using the formula: time difference = (longitude difference / 15) hours Using this formula, we have: longitude difference = 139°E - 121°E = 18°E time difference = (18°E / 15) hours = 1 hour and 12 minutes Answer: A. 1 hour and 12 minutes 76. Using the spherical law of cosines, we have: cos a = cos b cos c + sin b sin c cos A cos 46° = cos b cos 75° cos b = cos 46° / cos 75° Using a calculator, we get: cos b ≈ 0.5209 b ≈ 59.9° However, since this is a spherical triangle, we need to convert this to a spherical angle by using the formula: Spherical angle = 180° - angle in degrees Therefore, b = 180° - 59.9° = 120.1°. Answer: Not in options. 77. Using the spherical law of sines, we have: sin a / sin A = sin c / sin C sin 80° / sin 72° = sin 115° / sin 90° sin a = (sin 80° / sin 72°) sin 90° a = sin⁻¹ (sin 80° / sin 72°) Using a calculator, we get: a ≈ 96.2° Using the fact that the angles in a triangle add up to 180°, we can find angle C: C = 180° - A - a C = 180° - 72° - 96.2° C ≈ 11.8° Answer: Not in options. 78. The spherical excess of a spherical triangle is given by the formula: Excess = sum of angles - (n - 2) × 180° where n is the number of sides of the triangle. Since all angles of a right spherical triangle are 90°, we have: Excess = 90° + 90° + 90° - (3 - 2) × 180° Excess = 270° - 180° Excess = 90° Therefore, the spherical excess of the given spherical triangle is 90°. Answer: A. 45° 79. Using the distance formula, we have: distance = sqrt [(x2 - x1)² + (y2 - y1)²] distance = sqrt [(-2 - 4)² + (5 - (-3))²] distance = sqrt [(-6)² + 8²] distance = sqrt [36 + 64] distance = sqrt 100 distance = 10 Therefore, the distance between A and B is 10 units. Answer: C. 10 80. Since the points (a,1), (b,2), and (c,3) are collinear, we can use the slope formula to find the slope of the line passing through them: slope = (y2 - y1) / (x2 - x1) slope = (3 - 1) / (c - a) slope = 2 / (c - a) Since they are collinear, the slope must be the same for all pairs of points. Therefore, we have: 2 / (c - a) = 1 / (b - a) 2(b - a) = c - a Using the given choices, we can substitute in values for a, b, and c to see which one satisfies this equation: A. c - b = c - a B. c - b = b - a C. c - a = a - b D. c - a = b - a Trying option D, we have: c - a = b - a 2(b - a) = b - a b - a = 0 b = a Therefore, option D is true. Answer: D. c - a = b - a 81. Using the slope formula, we have: slope = (y2 - y1) / (x2 - x1) slope = (7 - 3) / (10 - (-5)) slope = 4 / 15 The inclination of a line is the angle it makes with the positive x-axis. We can find this angle using the tangent function: tan θ = slope θ = tan⁻¹ (slope) Using a calculator, we get: θ ≈ 14.93° Therefore, the inclination of the line passing through (-5,3) and (10,7) is approximately 14.93°. Answer: B. 14.93