Trigonometric Ratio Addition Formulae

Hello mathematicians, today’s topic on mathematics is trigonometric ratio addition formulae. Note that an ebook will be reveal on how to pass mathematics examination so consider yourself lucky.

trigonometric ratio addition formulae

Session Objectives

At the end of this session, candidates should be able to:
1) Use the addition formulae in solving simple trigonometric problems.

Addition Formula

trigonometric ratio of compound angles formulae

Consider the diagram above,

POX = A = QOP = B
QOX = POX + QOP
= A + B

Take a point C on OQ.
1)Draw a perpendicular from C to meet OP at D,

2) Draw a perpendicular from D to meet CF at G,

3) And lastly draw a perpendicular from D to meet OX at E

Note the element at the middle is equivalent to the angle e.g POX = 40° means O = 40° (I.e the triangle POX = 40°)

From the above diagram;
ODE = 90° – A

GDO = A since GDE = 90°

CDG = 90° – A since CDO = 90°,
hence GCD = A

Finding Expression For Sin(A + B), sin(A – B), cos(A + B), and cos(A – B) in terms of sinA, sinB, cosA and cosB

First Expression

Find sin( A ± B)

From the above diagram,
Sin(A ± B) = sin QOX

= CF
    OC
= CG + GF
OC
= CG+ GF
   OC   OC
= GF + CG
   OC    OC
= DE + CG
  OC   OC
= DE × OD
   OD   OC
= sinA × cosB
+ CG × CD
CD   OC
= cosA × sinB

hence;
Sin(A + B) = sinA.cosB + cosA.sinB
eqn (1)
Note : in mathematics . means multiplication

Substitute B to –B in equation (1)

Hence,

Sin(A – B) = sinA.cosB – cosA.sinB
eqn (2)

Second Expression

Find cos(A ± B)

cos(A + B) =

= OF
  OC
= OE – FE
OC
= OEFE
  OC   OC
= OEGD
  OC   OC
= OEOD
 OD   OC
= OE × OD
 OD   OC
= cosA × cosB
GD × CD
 CD   OC
= sinA × sinB
cos(A + B) = cosA.cosB – sinA.sinB
Substitute B to –B

Hence

cos(A – B) = cosA.cos( –B) – sinA.sin( –B)
cos(A – B) = cosA.cosB + sinA.sinB

Third Expression

Find tan(A ± B)
In mathematics tan

= Sin
    Cos
tan(A + B)

= sin(A + B)
cos(A + B)
= sinAcosB + cosAsinB
cosAcosB – sinAsinB
Divide both numerator and denominator by cosAcosB

First numerator expression
tan (A + B)

=     sinAcosB +  cosAsinB
      cosAcosB     cosAcosB
=   sinA + sinB
      cosA    cosB
Denominator expression
tan (A + B)

=  cosAcosB –    sinAsinB
    cosAcosB       cosAcosB
=  1 –   sinAsinB
             cosAcosB
tan (A + B)

=    tanA + tanB
       1 – tanAtanB
Substitute B to –B

tan (A – B)

=     tanA + tan( –B)
         1 – tanAtan( –B)
Hence,

tan (A – B)

=    tanA – tanB
      1 + tanAtanB
The formulae which we have expressed above are called Addition Formulae

note : Addition formulae are not only true for acute compound angles but are true for all compound angles.

Let look at some examples,

Example 1

Using addition formulae, evaluate, sin 75°, sin 255°, cos 195°, cos 15°, and tan 195°.

Solution 1

sin 75° = sin (30° + 45°)
= sin 30° + cos 45° + cos 30° + sin 45°
= 1 × √2 + √3 × √2
2
= ¼ × (√6 – √2)

Solution 2

cos 15° = cos (45° – 30°)
= cos 45° cos 30 + sin 45° sin30°
= √2 × √3+ √2 × 1
2
= ¼ × (√6 + √2)

Solution 3

sin 225° = sin (180° + 75°)
= sin 180° cos 75° + cos 180° sin 75°
= 0 – sin 75°
= – ¼ × (√6 + √2)

Solution 4

cos 195° = cos (180° + 15°)
= cos 180°cos15° – sin 180°sin 15°
= –cos 15° – 0
= – ¼ × (√6 + √2)

Solution 5

tan 195° = tan (180 + 15)

= tan 180° + 15°
1 – tan 180°tan15°
= 0 + tan15°
1 – 0
= tan 15°
3 – √3
3 + √3

Common Angles And Their Equivalents

1) sin 60 = cos 30

= 0.866 or √3
                   2
2) sin30 = cos 60 = 0.5 or ½

3)

sin 45 = cos 45 = 0.7071 = 1
                                            √2
4) tan 60 = 1.732 = √3

5)

tan 30 = 0.577 = 1
                            √3

Summary

Addition Formulae
1) sin (A + B) = sinAcosB + cosAsinB
2) sin (A – B) = sinAcosB – cosAsinB
3) cos (A + B) = cosAcosB – sinAsinB
4) cos (A – B) = cosAcosB + sinAsinB
5) tan (A + B)

=  tanA + tanB
     1 – tanAtanB

6) tan (A – B)

=    tanA – tanB
       1 + tanAtanB

Here is the goldmine, do you have gsce examination ahead and you are worried about the examination, here is an ebook that explain all you need to pass the examination note that, the ebook is not only applicable to students having gsce examination, the ebook will do just fine for any mathematics examination.

Leave a Reply