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a cos x + b sin x = R cos(x - )

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In this unit we explore how the sum of two trigonometric functions, e.g. 3 cos x + 4 sin x, can

be expressed as a single trigonometric function. Having the ability to do this enables you to

solve certain sorts of trigonometric equations and find maximum and minimum values of some

trigonometric functions.

In order to master the techniques explained here it is vital that you undertake the practice

exercises provided.

After reading this text, and/or viewing the video tutorial on this topic, you should be able to:

? express the sum of two trigonometric functions, a cos x + b sin x, in the form R cos(x - ).

? use this technique to solve some equations.

? use this technique to locate maximum and minimum values.

Contents

1. Introduction 2

2. The graph of y = 3 cos x + 4 sin x 2

3. The expression R cos(x - ) 3

4. Using the result to solve an equation 4

5. Finding maximum and minimum values 7

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1. Introduction

In this unit we are going to have a look at a particular form of trigonometric function. Consider

the following function, which is a sum of two trigonometric functions:

3 cos x + 4 sin x

You will find that in some applications, for example in solving trigonometric equations, it is helpful

to write these two terms as a single term. We study how this can be achieved in this unit.

2. The graph of y = 3 cos x + 4 sin x

We start by having a look at the graph of the function y = 3 cos x + 4 sin x. This is illustrated in

Figure 1.

y

5

x

-5

Figure 1. A graph of the function y = 3 cos x + 4 sin x.

If you have a graphical calculator you should check that you can reproduce this graph for yourself.

(The calculator mode should be set to work in radians rather than degrees). Observe that the

maximum and minimum values of this function are 5 and -5 respectively. Also note that the

graph looks like the graph of a cosine function except that it is displaced a little to the right.

To emphasise this, in Figure 2 we show this function again, and also the graph of y = 5 cos x for

comparison.

y

5

x

-5

Figure 2. Graphs of y = 5 cos x and y = 3 cos x + 4 sin x.

In fact the function 3 cos x + 4 sin x can be expressed in the form 5 cos(x - ) where is an angle

very close to 1 radian. It is the presence of the term which causes the horizontal displacement.

In the following section we will see how the more general expression a cos x+b sin x can be written

as R cos(x - ). In the example above note that the three numbers appearing in the problem,

i.e. 3, 4 and 5 form a Pythagorean triple (i.e. 32 + 42 = 52). This will be true more generally:

we will see that R2 = a2 + b2.

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3. The expression R cos(x - ).

We study the expression R cos(x-) and note that cos(x-) can be expanded using an addition

formula.

R cos(x - ) = R(cos x cos + sin x sin )

= R cos x cos + R sin x sin

We can re-order this expression as follows:

R cos(x - ) = (R cos ) cos x + (R sin ) sin x

So, if we want to write an expression of the form a cos x + b sin x in the form R cos(x - ) we

can do this by comparing

a cos x + b sin x with (R cos ) cos x + (R sin ) sin x

Doing this we see that

a = R cos (1)

b = R sin (2)

How can we use these to find values for R and ? By squaring each of Equations (1) and (2)

and adding we find

a2 + b2 = R2 cos2 + R2 sin2

= R2(cos2 + sin2 )

= R2

since cos2 + sin2 is always 1.

Hence

R = a2 + b2

It is conventional to choose only the positive square root, and hence R will always be positive.

What about the ?

We can find by dividing Equation (2) by Equation (1) to give

R sin = b

R cos a

so that

b

tan = a

Knowing tan we can find . So, now we can write any expression of the form a cos x + b sin x

in the form R cos(x - ).

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How do you turn Sin into COS? sine function can be changed to cosine and vice versa by adding 90 degrees and its multiples in domain of function so. Sin (a+90)= cos a it is +ve as in angle lies in 2nd quad if a is less than 90 and sine is + ve in 2nd quad. 15K views. ยท.

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