This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the problems: a) (i) Distance AC Step 1: Find the angle BAC. The sum of angles in a triangle is 180^. BAC = 180^ - ABC - ACB BAC = 180^ - 79^ - 40^ BAC = 61^ Step 2: Use the Sine Rule to find the distance AC. (AC)/(( ABC)) = (AB)/(( ACB)) (AC)/((79^)) = 15 km(40^) AC = (15 × (79^))/((40^)) AC = (15 × 0.9816)/(0.6428) AC = (14.724)/(0.6428) AC ≈ 22.91 km The distance AC is 22.91 km. (ii) Area of triangle ABC Step 1: Use the formula for the area of a triangle given two sides and the included angle. Area = (1)/(2) × AB × AC × ( BAC) Using the value of AC ≈ 22.906 km from the previous calculation for better precision. Area = (1)/(2) × 15 km × 22.906 km × (61^) Area = (1)/(2) × 15 × 22.906 × 0.8746 Area = 7.5 × 22.906 × 0.8746 Area = 171.795 × 0.8746 Area ≈ 150.25 km^2 The area of triangle ABC is 150.25 km^2. (iii) Shortest distance from B to AC Step 1: The shortest distance from point B to line AC is the perpendicular height from B to AC. Let this height be h. We can use the formula h = AB ( BAC). h = 15 km × (61^) h = 15 × 0.8746 h ≈ 13.12 km The shortest distance from B to AC is 13.12 km. b) Solve the equation = 0.937 for 0^ ≤ ≤ 360^. Step 1: Find the principal value of . _1 = ^-1(0.937) _1 ≈ 20.46^ Step 2: Find the second solution. Since is positive, the solutions lie in the 1st and 4th quadrants. The second solution is: _2 = 360^ - _1 _2 = 360^ - 20.46^ _2 = 339.54^ The solutions are 20.46^, 339.54^. c) Sketch the graph of y = for 0^ ≤ ≤ 360^. To sketch the graph of y = , plot the following key points and draw a smooth curve through them: • At = 0^, y = (0^) = 0. • At = 90^, y = (90^) = 1. • At = 180^, y = (180^) = 0. • At = 270^, y = (270^) = -1. • At = 360^, y = (360^) = 0. The graph starts at (0^, 0), rises to a maximum of 1 at (90^, 1), falls to 0 at (180^, 0), continues to fall to a minimum of -1 at (270^, -1), and rises back to 0 at (360^, 0). That's 2 down. 3 left today — send the next one.