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Year 10
Higher

Applying trigonometric ratios in 3D

I can apply knowledge of trigonometric ratios to 3D problems.

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Share activities with pupils
New
New
Year 10
Higher

Applying trigonometric ratios in 3D

I can apply knowledge of trigonometric ratios to 3D problems.

Download all resources
Share activities with pupils

Lesson details

Key learning points

  1. If angles are involved, then you may wish to apply trigonometric ratios to the 3D problem
  2. This allows you to calculate the angle between the line and the plane
  3. It can also be used to find missing lengths if the angle is known

Common misconception

Trigonometric ratios cannot be applied to 3D shapes.

Pupils may benefit from imaging the cuboid is 'sliced' through and a triangle drawn on the now visible face.

Keywords

  • Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.

  • Trigonometric ratios - The trigonometric ratios are ratios between each pair of lengths in a right-angled triangle.

  • Cross-section - A cross-section is a 2D face made from cutting straight through any plane of a 3D object.

Pupils may benefit from using the frame of a cuboid and some string so they can physically construct the right-angled triangles within it.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.
A is a 3D shape with 6 faces, where there are 3 pairs of congruent rectangles. It also has 8 vertices and 12 edges.
Correct Answer: cuboid
Q2.
For which of these triangles, would you use the cosine ratio, to calculate $$x$$?
An image in a quiz
An image in a quiz
Correct Answer: An image in a quiz
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An image in a quiz
Q3.
Which of these calculations are correct to find the length of $$x$$?
An image in a quiz
$$x=5.5\sin(42^\circ)$$
$$x=5.5\cos(42^\circ)$$
Correct answer: $$x=5.5\tan(42^\circ)$$
Q4.
Work out $$x$$ to 1 decimal place.
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Correct Answer: 26.7, 26.7 cm
Q5.
Work out $$x^\circ$$ to the nearest degree.
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Correct Answer: 32
Q6.
Work out the base angle in this isosceles triangle, to the nearest degree.
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Correct Answer: 69

Exit quiz

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6 Questions

Q1.
This 3D shape is a cuboid. Work out the length of the dashed line marked, $$x$$, to 1 decimal place.
An image in a quiz
Correct Answer: 21.5 cm, 21.5
Q2.
This 3D shape is a cuboid. Work out the length of the line that the arrow is pointing at (the diagonal of the bottom face), to 1 decimal place.
An image in a quiz
Correct Answer: 17.4 cm, 17.4
Q3.
Which of these calculations are correct for working out the volume of this cuboid?
An image in a quiz
$$18\times19\times34$$
$$18\times19\times19\sin(34^\circ)$$
$$18\times19\times19\cos(34^\circ)$$
Correct answer: $$18\times19\times19\tan(34^\circ)$$
Q4.
Work out the length of the edge marked $$x$$ on this triangular prism. Give your answer to 3 significant figures.
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Correct Answer: 7.51 cm, 7.51
Q5.
The volume of this triangular prism is $$\text{ cm}^3$$, to 1 decimal place.
An image in a quiz
Correct Answer: 420.4
Q6.
Which of these calculations are correct to find the volume of this right pyramid?
An image in a quiz
$$\frac{1}{3}\times10\times14\times \frac{5}{\cos(64^\circ)}$$
$$\frac{1}{3}\times10\times14\times \frac{10}{\cos(64^\circ)}$$
Correct answer: $$\frac{1}{3}\times10\times14\times 5\tan(64^\circ)$$
$$\frac{1}{3}\times10\times14\times 10\tan(64^\circ)$$