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 to your trigonometry problems: 6. A rhombus has diagonals of length 12 cm and 7 cm respectively. Find the larger angle of the rhombus. Step 1: Understand the properties of a rhombus. The diagonals of a rhombus bisect each other at right angles. This forms four congruent right-angled triangles. The half-diagonals are the legs of these right-angled triangles. Half of diagonal 1: x = 12 cm2 = 6 cm. Half of diagonal 2: y = 7 cm2 = 3.5 cm. Step 2: Find the angles within one of the right-angled triangles. Let 2 be the larger angle of the rhombus and 2 be the smaller angle. The diagonals bisect these angles. Consider one of the right-angled triangles formed by the half-diagonals. The angle is opposite the longer half-diagonal (x) and adjacent to the shorter half-diagonal (y). () = oppositeadjacent = (x)/(y) () = (6)/(3.5) () ≈ 1.7143 Step 3: Solve for and the larger angle. = (1.7143) ≈ 59.74^ The larger angle of the rhombus is 2. 2 ≈ 2 × 59.74^ 2 ≈ 119.48^ The larger angle of the rhombus is approximately 119.48^. 10. From a point A which is 60 metres from the base of a building B and on the same level, the angle of elevation to the top of the building C is 51^ and to the top of the flagpole D on top of the building the angle of elevation is 56^. Find the height of the flagpole. Step 1: Define variables and set up trigonometric relationships. Let AB be the distance from point A to the base of the building B, so AB = 60 m. Let h_C be the height of the building (BC). Let h_D be the total height from the base of the building to the top of the flagpole (BD). Let h_F be the height of the flagpole (CD). The angle of elevation to the top of the building is CAB = 51^. The angle of elevation to the top of the flagpole is DAB = 56^. Step 2: Calculate the height of the building (h_C). Using the right-angled triangle ABC: ( CAB) = (BC)/(AB) (51^) = (h_C)/(60 m) h_C = 60 m × (51^) h_C ≈ 60 m × 1.2349 h_C ≈ 74.094 m Step 3: Calculate the total height to the top of the flagpole (h_D). Using the right-angled triangle ABD: ( DAB) = (BD)/(AB) (5