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Trigonometry: Sum and Product of Sine and Cosine

On this page, we look at examples of adding two ratios, but we could go on and derive relationships for more than two.
Trigonometry Contents

Page Contents

  1. Sum of Sine and Cosine
    1. Sines: sumSines1.gif
    2. Cosines (Sum) sumCosPlus.gif
    3. Cosines Difference:
    4. Sum of Cosine and Sine: sumCosAndSin.gif
  2. Products



Sum of Sine and Cosine

Sines Sum

We wish to show that
sumSines1.gif [1.1]

In particular, we wish to show that
sumSines2.gif [1.2]

Let
A=(x+y) [1.3]
B=(x-y)  [1.4]

So, by adding and subtracting Equations 1.3, 1.4 we find:
A+B=2x, so:
x=(A+B)/2 [1.5], and
A-B=2y, so:
y=(A-B)/2  [1.6]

Substituting A and B for x and y, we find:
sin x+sin y=sin ( (A+B) )/2+sin( (A-B)/2 ) [1.7]

Using the compound angle formulae with each part:
sin ( (A+B) )/2=sin A/2 ·cos B/2 +cos A/2·sin B/2 (i)
And
sin( (A-B)/2 )=sin A/2 ·cos B/2 − cos A/2·sin B/2 (ii)

Adding (i) and (ii) we find:
sin ( (A+B) )/2 + sin( (A-B)/2 ) =2 · sin A/2 · cos B/2 [1.7b]

Because (A+B)/2=x [From 1.5] and (A-B)/2=y [From 1.6], substituting these in 1.7b, we find the relationship below:

sumSines2.gif [1.2]

Sines Difference

In a similar fashion:
sumSines3.gif
Substituting from Equations 1.3 and 1.4, that is:
A=(x+y) [1.3 repeated]
B=(x-y)  [1.4 repeated]
We find:
sin x−sin y=sin (A+B)−sin(A−B) [1.7]
Using the compound angle formulae with each part:
sin ( (A+B) )/2=sin A/2 ·cos B/2 +cos A/2·sin B/2 (i)
And
sin( (A-B)/2 )=sin A/2 ·cos B/2 − cos A/2·sin B/2 (ii)

Subtracting ii from i
sin ( (A+B) )/2 − sin( (A−B)/2 )=
2 ·cos A/2·sin B/2
 (because two terms cancel) (iii)
Substituting in (iii), the relationships (A+B)/2=x [From 1.5], (A-B)/2=y,  
A/2=(x+y)/2, and B/2=(x-y)/2 we find the relationship below:


sumSines3.gif [1.8]

Cosines Sum

The formulae for the sum of two cosines and for the difference are a little different (The addition is in terms of cosines: the substraction in terms of sines).

We wish to show that:sumCosPlus.gif [2.1]

Let A=(x+y)/2 [2.2], and
B=(x-y)/2  [2.3]
So, x=A+B, and y=A-B [2.4]

And
cos x+cos y=cos(A+B)+cos(A-B)

Expanding the right-hand side using the compound angle formula:
cos(A+B)+cos(A-B)=cosA·cosB-sinA·sinB+cosA·cosB+sinA·sinB
=2·cosA·cosB

Using Equations 2.2 and 2.3 to convert the A and B back to x and y:
sumCosPlus.gif
which is Equation 2.1, the result we sought.




Cosines Difference

The formula for the difference between two cosines is: [3.1]
To prove this, Let A=(x+y)/2, and B=(xy)/2  [2.3, repeated]
So, x=A+B, and y=AB [2.4, repeated]
By substituting [2.4]  in the left-hand side of Equation 3.1, we get:
cos x−cos y=cos(A+B)−cos(AB)

Expanding the compound angles, using the compound angle formulae:
cos(A+B)−cos(AB)=cosA·cosB−sinA·sinB−(cosA·cosB+sinA·sinB)
=−2·sin [(x+y)/2]·sin[(xy)/2]
Noting that −sin (θ)=sin (-θ), we can write −sin[(xy)/2]=sin[(y-x)/2] to remove the minus sign. And substituting for A and B [using 2.3 above], we get our equation:

which is Equation 3.1.

Sum of Cosine and Sine

The sum of the cosine and sine of the same angle, x, is given by:sumCosAndSin.gif [4.1]
We show this by using the principle cos θ=sin (π/2−θ), and convert the problem into the sum (or difference) between two sines.

We note that sin π/4=cos π/4=1/√2, and re-use cos θ=sin (π/2−θ) to obtain the required formula. 

Sum


The plus option gives:
cosPlusSign.gif [4.2]

We can write cos x as sin (π/2−x), so the left-hand side of Equation 4.2 becomes:
=sin (π/2−x)+sin x [4.3]

Which is the sum of two sines. Using the formula for the sum of two sines (above):
sumSines2.gif [1.2, repeated]

We get, by substituting in Equation 4.3:
cos x+ sin x=2·sin(π/4)·cos(π/4-x)

Noting sin π/4=cos π/4=1/√2:
cos x+ sin x=2/√2·cos(π/4-x)

Noting that: cos(π/4-x)=sin(π/2-(π/4-x))=sin (π/4+x)  [cos θ=sin (π/2−θ)]

We have:
cosPlusSign.gif
That is, Equation 4.2, which we wished to prove.

Difference

cosSineDifference.gif [5.1]
Using the same form as before...

We can write cos x as sin (π/2−x), so the left-hand side of Equation 5.1 becomes:

=sin (π/2−x)−sin x [5.2]

Which is the difference of two sines. Using the formula for the sum of two sines (above):
sumSines3.gif  [repeated]

We get, by substituting in Equation 5.2:
cos x- sin x=2·sin(π/4-x)·cos(π/4)

Noting sin π/4=cos π/4=1/√2:
cos x- sin x=2/√2·sin(π/4-x)

That is:
cosSineDifference.gif
That is, Equation 5.1, which we wished to prove.

Products

productSines.gif [6.2]
productCosines.gif [6.3]
productCosSine.gif [6.4]

We can show these relationships are true by expanding the right-hand sides using the compound angle formulas, the result occurs immediately.  




Trigonometry Contents

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