MHF 4U3 – ADVANCED FUNCTIONS Unit 2 – Trigonometry

Lesson 39 - Compound Angle Identities

Consider the problem of expressing a function such as cos(A + B) as a function of the angles A and B separately. To begin with, observe carefully that the cosine of a sum is not equal to the sum of the cosines. Example 1: Does cos(30° + 45°) = cos30° + cos45°? Solution: cos(30° + 45°) = cos75° = 0.2588

cos30° + cos45° = 21

23+

= 223 +

= 1.5731 Clearly cos(30° + 45°) ≠ cos30° + cos45°.

So what is the correct identity for cos(A + B)? To prove the addition formula for cos(A + B), start by taking three points on a unit circle as shown, where ∠QOR= A and ∠ROS = B. The coordinates of Q(x, y) can be found as follows:

rxA =cos but r = 1 (unit circle), therefore x = cosA

O

Q

S

R A B

Similarly, ryA =sin but r = 1 (unit circle), therefore y = sinA

Thus point Q(x, y) is Q(cosA, sinA). The coordinates of S(x, y) can be found as follows:

rxB =cos but r = 1 (unit circle), therefore x = cosB

Similarly, ryB =sin but r = 1 (unit circle), therefore y = sinB. But in quadrant IV,

y is negative. Thus point S(x, y) is S(cosB, -sinB). The length of the line segment QS can be calculated as follows:

QS = ( ) ( )22 sinsincoscos BABA ++− If we rotate the figure (as shown), the length of QS (now called Q’S’) must remain the same. The coordinates of Q’(x, y) can be found as follows:

( )rxBA =+cos but r = 1 (unit circle), therefore x = cos(A + B). Notice that we do

not need to worry about the sign in this case. Cosine will correspond (in sign) to that of x depending on the value of the angle (A + B).

Similarly, ( )ryBA =+sin but r = 1 (unit circle), therefore y = sin(A + B)

Thus point Q’(-x, y) is Q’(cos(A + B), sin(A + B)). The length of the line segment QS can be calculated as follows: Q’S’ = ( )( ) ( )( )22 0sin1cos −++−+ BABA

O

Q’

S’ (1, 0)

R’

A B

Since QS = Q’S’, we can equate the two expressions.

( ) ( )22 sinsincoscos BABA ++− = ( )( ) ( )( )22 0sin1cos −++−+ BABA ( ) ( )22 sinsincoscos BABA ++− = ( )( ) ( )BABA ++−+ 22 sin1cos

BBAABBAA 2222 sinsinsin2sincoscoscos2cos ++++− = ( ) ( ) ( )BABABA ++++−+ 22 sin1cos2cos BABA sinsin2coscos22 +− = ( )BA+− cos22 BABA sinsincoscos1 +− = ( )BA+− cos1 BABA sinsincoscos +− = ( )BA+−cos BABA sinsincoscos − = ( )BA+cos For the next examples you need to know the Reflection Identities sin (–C) = –sin C cos (–C) = cos C Example 2: Prove the identity 𝑐𝑜𝑠 !

!− 𝐶 = sin𝐶.

Solution: cos !

!− 𝐶 = cos !

!+ −𝐶

= cos !!cos −𝐶 − sin !

!sin −𝐶

= 0 cos −𝐶 − 1 − sin𝐶 = sin𝐶 If we replace C with !

!− 𝐶  we get the related identity:

cos !

!− !

!− 𝐶 = sin !

!− 𝐶

cos𝐶 = sin !!− 𝐶

Example 3: Prove the addition identity sin 𝐴 + 𝐵 = sin𝐴 cos𝐵 + cos𝐴 sin𝐵 Solution: Let C = A + B then

sin 𝐴 + 𝐵 = cos !!− 𝐴 + 𝐵

= cos !!− 𝐴 + −𝐵

= cos !!− 𝐴 cos −𝐵 − sin !

!− 𝐴 sin −𝐵

= sin𝐴 cos𝐵 − cos𝐴 − sin𝐵 = sin𝐴 cos𝐵 + cos𝐴 sin𝐵

Example 4: Prove cos 2𝐴 = cos𝐴 ! − sin𝐴 ! Solution: Let 2A = A + A then

cos 2𝐴 = cos 𝐴 + 𝐴 = cos𝐴 cos𝐴 − sin𝐴 sin𝐴 = cos𝐴 ! − sin𝐴 !

Example 5: Prove sin 2𝐴 = 2 sin𝐴 cos𝐴 Solution: Let 2A = A + A then

sin 2𝐴 = sin 𝐴 + 𝐴 = sin𝐴 cos𝐴 + cos𝐴 sin𝐴 = 2 sin𝐴 cos𝐴

Example 6: If 53cos =A ,

20 π

<< A , find ⎟⎠

⎞⎜⎝

⎛ +6

sin πA .

Solution: rxA =cos , therefore 3=x and 5=r

222 ryx =+ 259 2 =+ y

162 =y 4±=y

But A is in quadrant I therefore 4=y and 54sin =A

6sincos

6cossin

6sin πππ AAA +=⎟

⎠

⎞⎜⎝

⎛ +

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛ +21

53

23

54

6sin πA

103

1034

6sin +=⎟

⎠

⎞⎜⎝

⎛ +πA

10334

6sin +

=⎟⎠

⎞⎜⎝

⎛ +πA

There are several identities that you need to know. They are divided into 7 categories: Reciprocal Identities Quotient Identities Pythagorean Identities

AA

sin1csc =

AAA

cossintan = 1cossin 22 =+ AA

AA

cos1sec =

AAA

sincoscot = AA 22 tan1sec +=

AA

tan1cot = AA 22 cot1csc +=

Addition Identities Double Angle Identities ( ) BABABA sinsincoscoscos −=+ AAA cossin22sin = ( ) BABABA sinsincoscoscos +=− AAA 22 sincos2cos −= ( ) BABABA sincoscossinsin +=+

AAA 2tan1

tan22tan−

=

( ) BABABA sincoscossinsin −=−

( )BABABA

tantan1tantantan

−

+=+

( )BABABA

tantan1tantantan

+

−=−

Related Angle Identities ( ) AA sinsin =−π ( ) AA sin2sin −=−π ( ) AA coscos −=−π ( ) AA cos2cos =−π ( ) AA tantan −=−π ( ) AA tan2tan −=−π ( ) AA sinsin −=+π ( ) AA sinsin −=− ( ) AA coscos −=+π ( ) AA coscos =− ( ) AA tantan =+π ( ) AA tantan −=−

Corelated Angle Identities

AA cos2

sin =⎟⎠

⎞⎜⎝

⎛ −π AA cos

23sin −=⎟

⎠

⎞⎜⎝

⎛ −π

AA sin2

cos =⎟⎠

⎞⎜⎝

⎛ −π AA sin

23cos −=⎟

⎠

⎞⎜⎝

⎛ −π

AA cot2

tan =⎟⎠

⎞⎜⎝

⎛ −π AA cot

23tan =⎟

⎠

⎞⎜⎝

⎛ −π

AA cos2

sin =⎟⎠

⎞⎜⎝

⎛ +π AA cos

23sin −=⎟

⎠

⎞⎜⎝

⎛ +π

AA sin2

cos −=⎟⎠

⎞⎜⎝

⎛ +π AA sin

23cos =⎟

⎠

⎞⎜⎝

⎛ +π

AA cot2

tan −=⎟⎠

⎞⎜⎝

⎛ +π AA cot

23tan −=⎟

⎠

⎞⎜⎝

⎛ +π

Example 3: Prove that AAA 3sin4sin33sin −= . Solution: L.S. = ( )AA+2sin

= AAAA sin2coscos2sin + = ( ) AAAAAA sinsincoscoscossin2 22 −+ = AAAAA 322 sinsincoscossin2 −+ = ( ) ( ) AAAAA 322 sinsinsin1sin1sin2 −−+− = AAAAA 333 sinsinsinsin2sin2 −−+− = AA 3sin4sin3 − = R.S. Therefore L.S. = R.S. and the identity is true.

Homework: Supplementary Sheet 11 – Compound Angle Identities # 1 - 8