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Trigonometry - Compound Angles

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On this page, we claim to prove the sine and cosine relations of compound angles in a triangle, considering the cases where the sum of the angles is less than or more than 90°, and when one of the angles is greater than 90°
  1. Angle (α+β)<π/2
    1. Proof of the Sine and Cosine Compound Angles
      1. Proof of sin(α+β)=sinα cosβ +cosα sineβ
      2. Proof of cos(α+β)=sinα cosβ −cosα sineβ
  2. Angle (α+β)>π/2
    1. Proof of sin(α+β)=sinα cosβ +cosα sineβ, when α+β>π/2
    2. Proof of cos(α+β)=cosα cosβ +sinα sineβ, when α+β>π/2
  3. Angle (α+β)>π/2, and β>π/2
    1. Proof of sin(α+β)=sinα cosβ +cosα sinβ, when α+β>π/2, and β>π/2
    2. Proof of cos(α+β)=cosα cosβ − sinα sinβ, when α+β>π/2, and β>π/2

Angle (α+β)<π/2

The diagram below is composed of two right angled triangles, (OAB and OAE). The angle EOA is called α, and the angle AOB,  β.  The sum , α+β<90°. BD is perpendicular to OE and AX is perpendicular to BD, meeting it at X.
We note the following angles in the diagram (using "∠" for angle):
∠XAO=α (EOA and XAO are alterate angles of parallel lines, XA and OE)
∠XAB = 90- (OAB is a right angle, and XAB is 90°-α)
So angle ABX=α (triangle XAB is a right angled triangle, so ABX=90°-(90°-α)=α)
Also, ∠OBA in triangle ABO is (90°-β), so angle OAB=90°-α-β
Angles of interest are marked on the diagram (Fig 1).

Proof of the Sine and Cosine Compound Angles

Proof of sin(α+β)=sinα cosβ +cosα sineβ

We wish to prove that:
sin A+B
Or perhaps discover a relationship for the angle sum less than π/2
compoundAngle1
From the diagram above, we note:
sinab1[2.0], by the definition of sine.
We wish to obtain an expression for BD, and note that:
sinab2[2.01] (because both are segments of the line BD)

From triangles XAB, and OAE, BX=BA·cosα, and XD=OA·sinα, and substituting these values in 2.01, we obtain:
sinab3 [2.02]
Dividing this (2.02) by OB, we obtain (because sin(α+β)=BD/OB, equation 2.0)
sinab4 [2.03]
By definition of sine and cosine and from triangle OAB, we obtain:
sinab5 [2.04]
Substituting these values in 2.03, we obtain the required relationship:
sinab6 ■ [2.05]

Proof of cos(α+β)=sinα cosβ −cosα sineβ

(Fig 1 is repeated below. See above for the explanation of the angles)compoundAngle1
From the triangle OBD, we note that for angle OBD:
cosab2[2.06]
Recalling that sin(90-A)=cos(A):
cosab1 [2.07]
Hence,
cosab3 [2.08]
From the diagram we note:
cosab4 (Because OE is a straight line) [2.09]


From ΔOAB, we find:
cosagb (From the definitions of sine and cosine) [2.10]
Substituting this value for OD in equation 2.08, we have:
cosab6 [2.11]
From ΔOAB:
cosab7  (Definitions of sine and cosine) [2.12]
And substituting this in equation 2.11:
cosab8  [2.13] 
Which is the cosine of the sum of two angles in the acute angled triangle ODB. ■

Angle (α+β)>π/2

In Fig 2, ∠EOA=α and ∠AOB=β, and the sum α+β is greater than 90°.
BA is perpendicular to OA and the line BXD is perpendicular to the line DOE.
Line AX is perpendicular to BD.
Because angle OAB in ΔOAB is a right angle, then angle OBA=90°−β (as marked in the diagram)
In ΔODB, ∠DOB=180°−(α+β) (Angle of a straight line is 180°). And ∠DBO=90°-(180°−(α+β) )=(α+β)−90° (as marked in the diagram).
So, angle DBA=α (the sum of angles DBO and OBA)
Compound Angle 2

Proof of sin(α+β)=sinα cosβ +cosα sineβ, when α+β>π/2

In Fig 2, we note, by the definition of cosine:
sin22ab1 [3.01]
Because for any angle θ, cos(θ-π/2)=cos(-(π/2-θ)=cosθ, equation 3.01 becomes:
sin22ab2 [3.02]
sin22ab3 [3.03]
sin22ab4[3.04]
From equation 3.01, we obtain:
sin22ab6[3.05]
Now,
sin22ab6 (Segments of the line BD[3.06]
Using the definitions of sine and cosine in triangles ABX and OBD:
sin22ab7 [3.07]
Substituting BD in equation 3.05
sin22ab8 [3.08]
From triangle OAB and using the relationship, cos(π/2-β)=sin(β), we find:
sin22ab9 [3.09]
And
sin22ab10 [3.10]
Substituting these values in 3.08 and simplifying, we find:
sin22ab11 [3.11]
By starting the values with α (for presentation):
sin22ab12 ■ [3.12]

Proof of cos(α+β)=cosα cosβ +sinα sineβ, when α+β>π/2

[Fig 2 is repeated below for convenience.]
 Compound Angle 2

sin2ab2, definition of the sine [4.01]
Also from ΔDBO, we note for the angle DBO
sin2ab1[4.02]
Because sin(π/2-θ)=cosθ, and because the cosine of a negative angle is positive.
Hence:
sin2ab3, from 4.02 and 4.01 [4,03]
From the diagram we can find DO
sin2ab5, segments of the line DE [4.04]
From the definitions of the sine and cosine in ΔDBO ΔOBA
sin2ab4 [4.05]
Substituting in equation 4.03, we find:
sin2ab7 [4.06]

We also note that in ΔOBA, the following relationships hold:
sin2ab6, definition of sine and cosine [4.07]
Substituting these values in 4.06
sin2ab8 [4.07]
Expanding 4.07, we have:
sin2ab9 [4,08]
And rearranging to put alpha in the first positions (for cosmetic reasons only), we have
sin2ab10■  [4.09]



Angle (α+β)>π/2, and β>π/2

In Fig 3, below, angle POF=α, and angle FOB =β
BX and EA are parallel and perpendicular to DEO, which is parallel to XA.
BA is a line drawn perpendicular to FA from B
The angle AOD=α (Opposite angles). OAE=(π/2-α) (Angle sum of ΔOAE), so ∠BAE=α because it equals π/2−(π/2-α)
Angle OQB=(π/2+α) because it is the external angle of ΔAEQ, and is the sum of QEA (π/2) and EAQ(α).
In triangle OQB, the external angle POB (α+β)=π/2+α +∠QBO, hence, ∠QBO=β−π/2
∠DBQ=the external angle OQB (π/2+α)−QDB (π/2)=α
Finally, we note that ∠OBD=α+β−π/2
cos(α+β−π/2)=cos(−(π/2−(α+β))=cos((π/2−(α+β)) =sin(((α+β))
sin(α+β−π/2)=sin(−(π/2−(α+β))=−sin((π/2−(α+β)) =−cos(((α+β))

compoundAngle3

Proof of sin(α+β)=sinα cosβ +cosα sinβ, when α+β>π/2, and β>π/2

In Fig 3, we note that:sin331 [5.01]
And the cosine is:
sin332 [5.02]
Rearranging the cosine to the form cos(π/2-(α+β)
sin333 [5.03]
Hence, from 5.02, we have
sin334 [5.04]
Because
sin335 [5.05]
We have from ΔABD, using the definitions of sine and cosine:
sin336 [5.06]
Dividing by BO throughout, and noting that DB/BO=sin(α+β) from 5.02
sin337 [5.07]
That is, (omitting the first equality)
sin338 [5.08]
From ΔABO, and the definitions of sine and cosine, and the formula for complementary angles for sines and cosines[sin(π/2-θ)=cos(θ), cos(π/2-θ)=sin(θ)]
sin339 [5.09]
sin3310 [5.10]
Substituting the values for AB/BO and AO/BO in 5.08, we obtain the formula for the sine of a compound angle, when one is greater than π/2
sin3311 [5.11]
Which, by putting sin(α) first, for cosmetic reasons:
sin22ab12 ■

Proof of cos(α+β)=cosα cosβ − sinα sinβ, when α+β>π/2, and β>π/2

Figure 3 is repeated below. The angles are founds as before. compoundAngle3
In figure 3, we note that cos331 [6.01]
Using the the relationship between the sines and cosines of complementary angles:
cos332 [6.02]
Because for the general angle θ, sin(θ-π/2)=sin[-(π/2−θ)]=−sin(π/2−θ)=−cos(θ)

Hence, in ΔBDO
cos333[6.03]
We note for the line DEO=DE+EO, and because XA=DE, we have:
cos334 [6.04]
From ΔABX and ΔAOE and the definitions of cosine and sine
cos335 [6.05]
Dividing throughout by DO:
cos336 [6.06]
From 6.03, we have:
cos337 [6.07]
Hence, substituting this in 6.06
cos337b [6.07b]
From ΔABO
cos338 [6.08]
And also from ΔABO
cos339 [6.09]
Substituting these values in 6.07b:
cos3310 [6.10]
Multiplying throughout by minus one, and slightly rearranging:
cos3311 ■ [6.11]

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