Example 2

In this example, we will look at how to express product as sum of two trigonometry ratios.

 

Write the following as the sum of two trigonometry ratios.

1.       sin 19x cos 13x         2.    sin11α sin21α          3.    Cos 6p sin 4p          4.    Cos 13α cos 11α 5.    Sin 4x cos 9x

 

Solution

 

  1.  sin 19x cos 13x

From our product formula, sin and cos are connected by either sin X + sin Y or sin X – sin Y. The two when solved correctly will give us the same answer.

 

Using sin X + sin Y

 

sin X + sin Y = 2 sin (X + Y)/2 cos (X – Y)/2

divide both sides by 2

½ (sin X + sin Y) = sin (X + Y)/2 cos (X – Y)/2

 

We will relate our bolded part with the original question.

sin 19x cos 13x = sin (X + Y)/2 cos (X – Y)/2

 

equating sin

 

19x = (X + Y)/2

38x = X + Y --------------------------------- (equation 1)

Equating cos

 

13x = (X – Y)/2

26x = X – Y -------------------------- (equation 2)

 

From (1) and (2) above, eliminate y

X + Y = 38x

X – Y = 26x

2X      = 64x

X        = 64x/2

X        = 32x

 

Substitute 32x for X in (1)

X + Y = 38x

32x + Y = 38x

Y = 38x – 32x

Y = 6x

 

We will now substitute 32x and 6x for X and Y respectively in ½ (sin X + sin Y)

= ½ (sin X + sin Y)

= ½ (sin 32x + sin 6x)

∴ sin 19x cos 13x = ½ (sin 32x + sin 6x)

 

Also, if we decide to use sin X – sin Y

 

            Sin X – sin Y = 2 sin (X – Y)/2 cos (X + Y)/2

Divide both sides by 2

            ½ ( sinX – sinY) = sin (X – Y)/2 cos (X + Y)/2

            Equate the bolded to the original question

            sin (X – Y)/2 cos (X + Y)/2 = sin 19x cos 13x

 

            equate sin

            (X – Y)/2 = 19x

            X – Y = 38x --------------------------------------------------- (1)

             

            Equate cos

            (X + Y)/2 = 13x

            X + Y = 26x --------------------------------------------------- (2)

 

            Eliminate Y from (1) and (2)

            2X   =    64x

            X = 32x

            Substitute 32x for X in (2)

            X + Y = 26x

            32x + Y = 26x

            Y = 26x – 32x

            Y = - 6x

 

Substitute 32x and – 6x for X and Y respectively in ½ (sinX – sinY)

= ½ (sin X – sin Y)

= ½ (sin 32x – sin (-6x))

= ½ (sin 32x + sin 6x)

 

∴ sin 19x cos 13x = ½ (sin 32x + sin 6x)

Our answer is still the same as the first one above.

 

         2.     sin11α sin21α

Product of two sin can be obtained from cos X – cos Y

cos X – cos Y = -2 sin (X+Y)/2 sin (X – Y)/2

Divide both sides by – 2

            - ½ (cos X – cos Y) = sin (X+Y)/2 sin (X – Y)/2

            ½ (cos Y – cos X) = sin (X+Y)/2 sin (X – Y)/2

            Notice how the removal of minus affects the expression in the bracket.

            sin (X+Y)/2 sin (X – Y)/2 = sin11α sin21α

                        from the first sin

            (X+Y)/2 = 11α

            X + Y = 22α ------------------------------------- (1)

                        From second sin

            (X – Y)/2 = 21α

            X – Y = 42α ------------------------------------- (2)

 

            From (1) and (2) eliminate Y

            2X = 64α

            X = 32α

 

            From (1) sub 32α for X

            X + Y = 22α

            32α + Y = 22α

            Y = 22α - 32α

            Y = -10α

 

            Substitute for X and Y in ½ (cos Y – cos X)

            = ½ (cos Y – cos X)

            = ½ (cos (-10x) – cos 32x)

            = ½ (cos 10x – cos 32x)

 

            Note: cos(–x) is same as cos x

 

         3.     Cos 6p sin 4p

As we already know that the product of sin and cos can be obtained from either sin X + sin Y or sin X – sin Y, so I will use the latter to solve this question.

You can also use the former, we will get the same answer.

 

Sin X – sin Y = 2 sin (X – Y)/2 cos (X + Y)/2

Divide both sides by 2

            ½ ( sinX – sinY) = sin (X – Y)/2 cos (X + Y)/2

            Equate the bolded to the original question

            sin (X – Y)/2 cos (X + Y)/2 = Cos 6p sin 4p

          sin (X – Y)/2 cos (X + Y)/2 = sin 4p cos 6p

                   (X – Y)/2 = 4p

                        X – Y = 8p ------------------------------------- (1)

 

                        (X + Y)/2 = 6p

                        X + Y = 12p ------------------------------------ (2)

From (1) and (2) eliminate Y

 

                        2X = 20p

                        X = 10p

Substitute 10p for X in (2)

           

                        X + Y = 12p

                        10p + Y = 12p

                        Y = 12p – 10p

                        Y = 2p

Substitute 10p and 2p for X and Y respectively in ½ ( sinX – sinY)

= ½ (sinX – sinY)

= ½ (sin 10p – sin 2p)

 

∴ Cos 6p sin 4p = ½ (sin 10p – sin 2p)

 

        4.   cos 13α cos 11α

cos X + cos Y = 2 cos (X + Y)/2cos (X – Y)/2

½ (cos X + cos Y) = cos (X + Y)/2cos (X – Y)/2

cos (X + Y)/2 cos (X – Y)/2 = cos 13α cos 11α

 

(X + Y)/2 = 13α

X + Y = 26α ---------------------------------------------------- (1)

 

(X – Y)/2 = 11α

X – Y = 22α ---------------------------------------------------- (2)

 

From (1) and (2) eliminate Y

 

2X = 48α

X = 24α

 

From (1)  

X + Y = 26α

24α + Y = 26α

Y = 26α - 24α

Y = 2α

 

Substitute 24α and 2α for X and Y respectively in ½ (cos X + cos Y)

= ½ (cos X + cos Y)

= ½ (cos 24α + cos 2α)

 

∴ cos 13α cos 11α = ½ (cos 24α + cos 2α)  

              

      Exercise

Express the following as the sum of two trigonometric ratios.

1.  cos 3x cos 12x

2.  sin 13p cos5p

3.  sin 15α sin21α

4.  sin 9x cos 4x

5.  cos 36x sin 20x

6.  sin 27x sin 3x

7.  cos 5a cos 17a

8.  sin 8f cos 15f

 

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