The Sine Rule For A Level Maths
The Sine Rule For A Level Maths – Introduction
The sine rule is a mathematical principle used in trigonometry to relate the angles and sides of a triangle. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. In GCSE Higher Maths, the sine rule is introduced as a tool to solve problems involving non-right-angled triangles.
It allows students to find unknown side lengths or angles by using the ratios derived from the sine rule equation. At A Level Maths, the sine rule continues to be utilised in more advanced trigonometric applications, such as solving complex trigonometric equations and analysing the properties of triangles in three-dimensional space.
The history of the sine rule dates back to ancient times, with evidence of its usage found in the works of Greek mathematicians such as Euclid and Ptolemy. However, the exact origins and pioneers of the sine rule are not well-documented, making it challenging to provide precise dates for its first usage.
The Sine Rule Formula
This is the formula for the sine rule and it is recommended that you learn this formula:
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}This version of the formula is generally used to find a missing length.
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}This version of the formula is generally used to find a missing angle.
You can use either formula for any question that you encounter and we will look at a few examples throughout the remainder of this article.
The Sine Rule - Basic Examples
Example 1
In any question relating to triangles you need to help yourself by first of all drawing a sketch.
Example 2
Again to help answer the question you should draw a sketch of the triangle and one is shown below:
Here you want to find a missing angle and with all sine rule questions you need to determine which angle corresponds to which length. The missing angle is associated with the length of 3.8cm and so the sine rule can now be done as follows:
\begin{aligned} \frac{\sin C}{c} & =\frac{\sin A}{a} \\ \frac{\sin C}{3.8} & =\frac{\sin 35^{\circ}}{5.2} \\ \sin C & =\frac{3.8 \sin 35^{\circ}}{5.2} \\ C & =24.781 \ldots \\ B & =120^{\circ}(3 \text { s.f. }) \end{aligned}
The Sine Rule – Problem Solving
In order to be able to answer this question you again need to have a sketch and to be able to add the appropriate angles which you may need to find using various techniques such as using alternate angles.
Below is a sketch and it is detailed how the various angles have been found:
Using alternate angles and the facts that angles on a straight line add up to one hundred and eighty degrees.
\angle A C B=180^{\circ}-(80+35)^{\circ}=65^{\circ}To help find the missing angles you will need to recall some angles in parallel line facts that you were taught at GCSE. Remember all the information at GCSE is now expected knowledge at A Level. If there are any gaps in your knowledge then you need to take the appropriate steps to consolidate any gaps in knowledge.
Having this information you are now in a better position to answer the question.
Part a)
\begin{aligned} & \frac{A C}{\sin 80^{\circ}}=\frac{6}{\sin 65^{\circ}} \\ & A C=\frac{6 \sin 80^{\circ}}{\sin 65^{\circ}}=6.52 \mathrm{~km}(3 \text { s.f. }) \end{aligned}Part b)
\begin{aligned} & \frac{B C}{\sin 35^{\circ}}=\frac{6}{\sin 65^{\circ}} \\ & B C=\frac{6 \sin 35^{\circ}}{\sin 65^{\circ}}=3.80 \mathrm{~km}(3 \text { s.f. }) \end{aligned}The Sine Rule – Harder Examples
Here is the solution for this question:
Using the sine rule will give us the following:
\begin{aligned} & \frac{h}{\sin 9.5^{\circ}}=\frac{150}{\sin 62.5^{\circ}} \\ & h=\frac{150 \sin 9.5^{\circ}}{\sin 62.5^{\circ}}=27.9 \mathrm{~m} \end{aligned}
Within these examples you will have covered the necessary steps to be able to answer most questions. There will be questions that will require more knowledge that just the use of the sine rule. There will be questions where you could be expected to use the cosine rule as well as Pythagoras.
At A Level, being able to link to other topics is more of a challenge as you can also be expected to answer questions that still require GCSE knowledge but you are using say inverse trigonometric ratios.
However, one thing remains the same whether you are using the sine rule, cosine rule or even Pythagoras and that is to make sure you draw a clear diagram to help you visualise the question clearly.