Solving 3D Problems using Trigonometry

When teaching solving 3D problems using trigonometry we begin the lesson with a recap of Pythagoras Theorem and the three trigonometric ratios.  We do this by matching the ratio and equations to the respective right-angled triangle.

Students are encouraged to work in pairs and to show the diagrams as part of the working out on mini-whiteboards.   A nice way to differentiate this is to have the less able student do the writing.  I check to see whether they can correctly label the sides as this helps to know whether to use Sine, Cosine or Tangent.

Solving 3D Problems using Trigonometry

When solving 3D problems involving trigonometry, I demonstrate how to identify the right-angled triangle containing the unknown side or angle and sketch it out separately.  If there is not enough information given about this triangle, we look for further clues in another triangle involving the same line or angle.  Check out the video below for a quick demonstration.

Identifying the Right-Angled Triangles

To check the student’s understanding and progress we work through some examples from the Interactive Excel File on mini-whiteboards.

Students are asked to sketch out the necessary right-angled triangles needed to find the length FB and angle FBG.  Without explaining how, I ask them to calculate the length FB and show me their working on mini-whiteboards.  It is interesting that a couple of students try to apply the Cosine Rule to triangle FBG.   We feedback how to use Pythagoras’ Theorem to address the misconceptions.

Next, I ask students to use this new information to calculate angle FBG.  Some struggle to use the correct ratio but all students attempt to solve it as a right-angled triangle using either Sine, Cosine or Tangent.

I use the solving 3D problems using trigonometry Interactive Excel File to pose a different, but similar question for the class to attempt again on mini-whiteboards.  All students sketch the two relevant right-angled triangles.  Some continue to struggle calculating the angle using the correct trigonometric ratio but all students correctly calculate the length FB using Pythagroas’ Theorem.

Applying Trigonometry to Square Based Pyramids

The plenary challenges the class to identify the necessary right-angled triangles independently.   I provide a handout of the shape to some students so they can annotate the drawing.  After a couple of minutes of attempting the problem independently we stop the lesson to discuss different approaches.  This provides a starting point for some students while reassuring others.  The plenary problem normally takes between 10 to 12 minutes.  To end the lesson, I ask a willing student to demonstrate their working at the front of the class to their peers.  We address any misconceptions as they arise.

Revising 3D Trigonometry

Students revise how to use the Sine, Cosine and Tangent...
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Revising Trigonometry in Right-Angled Triangles

Students revise how to calculate unknown angles and lengths in...
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Revising the Cosine Rule

Students revise how to use the Cosine Rule to calculate...
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Exact Trigonometric Solutions

Students learn how to solve trigonometrical equations with exact solutions...
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