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.
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Here are the solutions for questions 3.2 and 3.3: 3.2 Determine the value of lengths y and z. Given a right-angled triangle XYZ with Y = 90^, Z = 30^, and side XY = 20. We need to find y (hypotenuse XZ) and z (side YZ). Step 1: Determine the length of y (hypotenuse XZ). We know the opposite side (XY = 20) and the angle Z = 30^. We use the sine function. ( Z) = OppositeHypotenuse (30^) = (XY)/(XZ) (1)/(2) = (20)/(y) y = 2 × 20 y = 40 Step 2: Determine the length of z (adjacent side YZ). We know the opposite side (XY = 20) and the angle Z = 30^. We use the tangent function. ( Z) = OppositeAdjacent (30^) = (XY)/(YZ) (1)/(sqrt(3)) = (20)/(z) z = 20sqrt(3) z ≈ 34.64 3.3 Determine the value of the following. Show all calculations. Given a right-angled triangle ABC with C = 90^, side AC = 6, and side BC = 8. Angle ABC = . Step 1: Calculate the length of the hypotenuse AB. Using the Pythagorean theorem: AB^2 = AC^2 + BC^2 AB^2 = 6^2 + 8^2 AB^2 = 36 + 64 AB^2 = 100 AB = sqrt(100) AB = 10 3.3.1 For angle (at B), the opposite side is AC and the hypotenuse is AB. = OppositeHypotenuse = (AC)/(AB) = (6)/(10) = (3)/(5) 3.3.2 ^2 + ^2 First, calculate . For angle , the adjacent side is BC and the hypotenuse is AB. = AdjacentHypotenuse = (BC)/(AB) = (8)/(10) = (4)/(5) Now substitute into the expression: ^2 + ^2 = ((3)/(5))^2 + ((4)/(5))^2 = (9)/(25) + (16)/(25) = (9+16)/(25) = (25)/(25) = 1 3.3.3 ( )/( ) Using the values calculated for and : ( )/( ) = (3)/(5)(4)/(5) = (3)/(5) × (5)/(4) = (3)/(4) What's next?