Sine Rule – more on the ambiguous case

Sin A / a    =    Sin B / b    =    Sin C / c

So you can use the Sine rule if you have two angles and one side (AAS):

Sin A / a    =    Sin B / x

Or if you have two sides and one angle (SSA):

Sin A / a    =    x / b

BUT remember that if you are given two sides and one angle of a triangle (SSA), you don’t have enough information to know exactly what that triangle looks like:


Look at the diagram.

Is it

Sin 35 / 6cm    =    Sin C / 10cm?


Sin 35 / 6cm    =   Sin D / 10cm? 

This is the ambiguous case (ambiguous means “could-be-either”). To identify the ambiguous case, you need to follow these rules:

  • If you have AAS, then everything is fine, go ahead and use the Sine rule.
  • If you have SSA and the side opposite the angle you have been given is the longer of the two sides you have been given, then everything is fine, go ahead and use the Sine rule.
  • Otherwise, you have an ambiguous case and you need to be careful.

What to do about the ambiguous case?

Remember that Sin(x) = Sin(180-x), for instance:

  • Sin(80) = Sin(100)
  • Sin(45) = Sin(135)
  • Sin(10) = Sin(170)
  • Sin(0) = Sin(180)

Your missing angle is either x or 180 – x. But…

And this is important…

When you use inverse sine to find your missing angle, it will always give you x rather than 180 – x.

Technically, it will always give you x such -90 <= x < 90.

Maths Test: Right Triangle Trigonometry and Sinusoidal Functions

I have decided to move your test to Wednesday 13th (Block E) and Thursday 14th (Block G). The test will cover the following areas:

  • Sinusoidal functions of the form A Sin (B(x – C)) + D and A Cos (B(x – C)) + D (in particular, could you model a periodic function with Sine and Cosine? How would they be different?)
  • Simple right-triangle trig: finding a missing side or a missing angle
  • Multi-step problems: finding a missing side or angle in order to find a further missing side or angle
  • Word problems, in which you need to construct a diagram first, and then find missing values
  • You will also need to be able to use trigonometric ratios to show whether or not a triangle exists. For instance, a triangle with angles of 30 degrees, 60 degrees and 100 degrees does not exist, neither does a triangle with sides 10cm, 5cm and 4cm.

The test will NOT include:

  • Area formula
  • Sine Rule
  • Cosine Rule

Revision resources are:

  • The diagram I prepared explaining the effect of A, B, C and D in A Sin (B(x – C)) + D
  • The materials in my GDrive shared documents for Block E and Block G 


Homework: using triangle trig

I’ve been surprised in both of my classes how many of you adopt a rather unsystematic approach to problem-solving. This week’s homework focuses on a neat, organised approach.

Your task is to do question 6 from the last handout. You may have already done it in class, but this task is all about presentation, so please do it again.

I would like to see:

  • Question number
  • Diagram
  • Statement of required precision in the final answer (significant figures, or decimal points if the question says so).
  • Intermediate results quoted with 2 extra significant figures.
  • Two statements outlining the method that you will use to solve the problem.
  • Clear working with intermediate results single-underlined and the final answer double-underlined.

I will check this work in class next week. It needs to be hole-punched and in your maths folder. If your work is incomplete, or scrappy, or you haven’t used a ruler, or you haven’t put it in your folder, then I will ask you to do it again during tutorials.

Here is an example:


Sheriff of Nottingham approach

I happened to speak with Mr S this afternoon and he made some comments about the Sheriff of Nottingham task that I think you should be aware of. He is expecting you to estimate the second zero (if you decide to use factored form) or the vertex* (if you use vertex form) and then to find a by using a data point (the position of Prince John). You will then need to check that you (a) clear the wall and (b) hit the target within a reasonably range of error (ie to the nearest metre).

There are other methods, and as long as they find the function, show that the wall is cleared and check that the target is hit, I will allow them. One analytical method is to choose another data point in addition to the known ones (0, 0) and (130, 10), and construct a system of three equations that you can solve simultaneously. The third data point would probably be something like (110, 60), ensuring that you clear the wall. This is very precise, but harder work. If you do it this way I will buy you the item of confectionery of your choice.**

* Don’t forget that if you know where both zeroes are, you can calculate the vertex, and if you know one zero and the vertex, you can calculate the other zero easily.

** Up to the value of 50PhP. Offer only open to my classes. Terms and conditions apply.