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3D Trigonometry Introduction

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3D Trigonometry Introduction

A survey team is checking a rooftop structure. Use right-angled triangles, Pythagoras’ theorem and trigonometric ratios to find lengths and angles in three dimensions. Give lengths to 2 decimal places and angles to 1 decimal place unless told otherwise.

Part A: Build the 3D picture

A right-angled triangle is often hidden inside a 3D shape. Identify the useful triangle before choosing a trigonometric ratio.

1.A cuboid has a floor that is 6 m long and 4 m wide. What is the length of the diagonal across the floor from the bottom-front-left corner to the bottom-back-right corner? Show the calculation.

2.A cuboid has a floor that is 6 m long and 4 m wide, and a height of 3 m. The cuboid’s space diagonal, its height and the floor diagonal form a right-angled triangle. Calculate the space diagonal.

3.A drone is directly above the cuboid’s bottom-front-left corner. It is 3 m high and 6 m along the length from that corner. Find the angle of elevation from the corner to the drone.

4.Choose the correct expression for the angle between a cuboid’s space diagonal and its floor diagonal. Let the floor diagonal be d and the space diagonal be s.

sin θ = 3 ÷ s
cos θ = d ÷ s
tan θ = s ÷ d
cos θ = 3 ÷ d

Part B: Apply the ratios

Use a clear sketch or name the right-angled triangle you use. Include units in your answers.

5.A rectangular storage container is 5 m long, 2 m wide and 2 m high. Find the length of its space diagonal.

6.A support cable runs from the ground to the top of a 4.8 m pole. Its ground anchor is 3.6 m from the pole’s base. Find the angle the cable makes with the ground.

7.A surveyor stands 12 m from the foot of a vertical mast. The angle of elevation from eye level to the top is 38°. How far above the surveyor’s eye level is the top of the mast?

8.A rectangular roof is 8 m long and 6 m wide. A straight brace runs from one bottom corner to the opposite top corner. Find the angle between the brace and the roof’s floor, assuming the vertical rise is 5 m.

9.True or false? In a cuboid, the space diagonal is always longer than each face diagonal. Explain your choice using the idea of perpendicular dimensions.

10.A cable is 13 m long and connects the top of a vertical mast to a ground anchor 5 m from the mast’s base. Find the mast’s height, then find the cable’s angle of elevation from the ground.