# Solving Problems with Non-Right-Angled Triangles

## Solving Problems with Non-Right-Angled Triangles

Solving problems with non-right-angled triangles involves multiple areas of mathematics ranging from  complex formulae to angles in a triangle and on a straight line.

As the GCSE mathematics curriculum increasingly challenges students to solve multiple step problems it is important for students to understand how to prove, apply and link together the various formulae associated to non-right-angled triangles.

## Proving the Sine, Cosine and Area Formulae for Non-Right-Angled Triangles

There are three ways of teaching students how to derive the Sine, Cosine and Area rules.

1. Work through each of the proofs with the students on the main whiteboard.  Students take notes of the steps involved and try it for themselves after my work has been erased.
2. Each step of derivation is jumbled up and the students reorder the stages to complete the proof. I use this approach the most.
3. Deriving each formula is set as a homework for students to teach it to themselves.  I have each of the derivations on YouTube for the Sine Rule, Cosine Rule and Area of a Triangle, Students have the time to demonstrate each proof during the starter of the relevant lesson.

## Sine Rule

I begin the topic of solving problems with non-right-angled triangles with the Sine Rule.  At the start of the lesson students arrange a jumbled up derivation using right-angled trigonometry.  In the main teaching phase we work through a series of problems involving missing angles and lengths.  The plenary is more challenging as students need to apply various angle properties to have a matching angle and side.

## Cosine Rule – Finding Lengths

After the Sine Rule we progress on to deriving and using the Cosine Rule to calculate unknown lengths.  The start of the lesson is another jumbled up proof for the students to complete.  When teaching the derivation most students are able to identify the a2 = b2 + c2 (-2bcCosA) as related to Pythagoras’ Theorem.  The development phase teaches students how to substitute known values into the formula.  As learning progresses students are challenged to combine the Sine and Cosine Rules within a single problem.

## Cosine Rule – Finding Angles In this lesson students learn how to find an unknown angle in a triangle when all the lengths are known.  We start with another jumbled up proof then quickly move on to sketching problems given as written descriptions.  Using mini-whitboards to sketch their diagrams helps students to visualise the correct information.

## Area of a Triangle

This is the final lesson in the topic.  Students apply both the Sine and Cosine rules to solve a range of problems involving the area of a triangle.  At the start students learn how to find the area of a triangle.  As learning progresses they use the area to calculate a missing angle or length.   Exam questions often include the area of a triangle in the non-calculator paper as Sin 30 can be worked out exactly without the need for a calculator.

## Solving problems involving multiple formulae Students struggle solving larger problems that involve multiple formulae as they do not take the time to devise a strategy.  I encourage students to take a minute or so to sketch a flowchart that will break the problem down into smaller, more manageable steps.

When the students have come up with a strategy we discuss how to identify which formula to use with the following prompts.

• Sine Rule – To be used when you have a matching pair of angle and side.
• Cosine Rule Length – To be used when a known angle is between two known lengths.
• Cosine Rule Angle – To be used when all three sides are known.
• Area Rule – To be used when the area is given or asked for.

When a problem is given without a diagram students find it difficult to visualise what they are being asked to calculate which is why an accurate sketch is so important.

When I teach solving problems with non-right-angled triangles students are required to sketch the diagrams as part of their working.  This helps them to accurately interpret the more wordy type problems.  Understanding the correct notation is important.  Angles are marked with capital letters.  Opposite sides are marked with lower case letters.

## Teach these lessons

### Pythagoras Theorem in 3D Shapes

Many problems involve three-dimensional objects or spaces.  Pythagoras’ Theorem can be used as much with these problems as those in plane shapes.

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