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C4 Trigonometry : The linear combination of sin and cos

Page history last edited by Steph Richards 15 years, 1 month ago

CONTRIBUTED BY "Atal" (GARETH WARD) who is a Year 13 Pupil at Hawthorn High School.

During C2, we came across questions such as:

 

Formula

 

Where we could use an identity to get everything in terms of cos x, and then find all values of X using CAST diagrams.

What if, however, We come across a question such as:

 

Formula

 

Where that isn't an option?

 

Welcome to C4.

 


 

 

a cos θ + b sin θ

 

The main goal here remains the same as it always has, we must break this equation down so that we can have θ = .....

In this case, a cos θ + b sin θ can be written as R cos(θ-α) or R sin(θ+α) , where R and α are integers to be defined.

 

Now, we must think back to the basics of trigonometry here, and what it all actually means. There must be a right angled triangle, where a and b determine the length of two sides. Therefore, we have the triangle:

 

Please excuse the cruddy MSPaint drawing, but that seemed the easiest way at the time of writing. 

 

Note that where we place the angle α is unimportant if we are simply asked to "Find all values of x" but it can become important for slightly different questions, as we will see later.

 

From this diagram, we can extract 4 facts.

 

1. With Pythagoras' theorem,

 

Formula

 

2, 3 and 4. Using trigonometry,

 

Formula

Formula

Formula

 

Now your probably thinking, "Thats great and all but why does that help us?"

 

Well, we can do this cheeky bit of mathematics, and take a factor out of a cos θ + b sin θ that was never there in the first place!

 

The factor we take out is the hypontenuse. We say that both a cos and b sin are being divided by h, as shown below;

 

Formula

 

Now, can you see the use of the triangle facts?

we've already proved that  

 

Formula

 

and

 

Formula

so this new equation:

 

Formula

 

Can be written as:

 

Formula

 

Now I hope you've read my Double and Compound Angles page!

 

If you think about it, we have ourselves a compound angle in the form Rcos(θ-α) only it has already been written in single angle form. This time around, we must reverse the formula. Doing so gives us the following:

 

Formula

 

Tada! We have ourselves an angle in the form R cos(θ-α), which C2 skills will easily fix up for you! Now lets try some actual questions, and throw something new into the mix.

 


 

Example Question

 

Write down 7 sin θ + 24 cos θ in the form Rsin(θ+α) where R and α are values to be determined.

 

This time around, where we place the angle α in our triangle is important, because only one of the 2 posibilities will give us the compound angle formula for sin.

We want part of it to eventually look like:

 

Formula

 

So that we can reverse it back to the form sin(θ+α) as the question asks us. Last time we had:

Which of course gave us the compund angle for cos. This time we are going to have:

as this reverses our facts, giving us

 

Formula 

 

and

 

Formula

 

Now, doing the same thing we did last time,

 

Formula

 

And subbing our values for a and b in (remember I am using the idea that a cos θ + b sin θ, so a = 24 and b = 7) we get:

 

Formula

 

Using our facts, we get:

 

Formula

 

and reversing our compound angle for sin, we get:

 

Formula

 

We are almost there! If you remember our 4th fact (also altered because we changed the position of α) ,

 

Formula

 

which when subbing in values for a and b and calculating it, gives us

 

Formula

 

We can simply sub it back into our compound angle, and we are done, the question is completed.

 

Formula

 

R is 25, α is 73.74.

 

They may also equate 7 sin θ + 24 cos θ to a value, lets say 3, and ask us to "find all values of θ between...." where we would simply do the following:

 

Formula

 

Use CAST for all values between 0, and 360 + 73.74 , as we will be taking that back off and we wouldnt want to miss any values. I would finish the question, but you've probably done it stacks of times by now if you are reading into C4 and since I made it up I doubt the numbers will be nice.

 

 There is only one more type of trigonometry question that you will be asked in this topic, and this is what I will cover next.

 


 

(to be completed)

 

Comments (4)

Steph Richards said

at 12:12 pm on Feb 1, 2009

Many thanks for such a comprehensive account. I am sure that the words of a student are far more powerful than the words of a teacher and that other students will thank you for this even more than I can, in the future.

Steph Richards said

at 12:15 pm on Feb 1, 2009

Can I ask where you managed to aquire the knowledge of the Latex code that allows you to write your formulae on the wiki?

Atal said

at 3:20 pm on Feb 1, 2009

I downloaded a program called MAthtype - If you copy anything you type in it, it will paste as latex

Luke said

at 5:18 pm on Sep 26, 2012

Only a 30 day trial is available for it, otherwise it's about £43 :/ what can you do on it once the trial expires? If you can still write into and copy/paste latex out of it, it's still worth getting.

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