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TRIGONOMETRIC FUNCTIONS OF SUM AND DIFFERENCE OF TWO ANGLES

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Trigonometric Functions of Sum and Differences Introduction

trig_sum_angle_1So far we have learnt basic trigonometric identities, in this section we will learn about trigonometric functions of the sum and difference of two angles using some basic results, following are the results which we must know before starting this topic:

1. Sin (-x) = - Sin x
2. Cos (-x) = Cos x
3. Sin (Π2 – x) = Cos x
4. Cos (Π2 – x) = Sin x
5. Sin (Π2 + x) = Cos x
6. Cos (Π2 + x) = - Sin x
7. Cos (Π – x) = - Cos x
8. Sin (Π – x) = Sin x
9. Cos (Π + x) = - Cos x
10. Sin (Π + x) = - Sin x
11. Cos (2Π – x) = Cos x
12. Sin (2Π - x) = - Sin x



Basic Proof

Consider the unit circle with centre at the origin.


trig_sum_angle_2Let x be the angle P4OP1 and y be the angle P1OP2. Then (x + y) is the angle P4OP2. Also let (– y) be the angle P4OP3.


Therefore, P1, P2, P3 and P4 will have the coordinates:

P1 (Cos x, Sin x),
P2 [Cos (x + y), Sin (x + y)],
P3 [Cos (– y), Sin (– y)] and
P4 (1, 0)

Consider the triangles P1OP3 and P2OP4. They are congruent


Therefore, P1P3 and P2P4 are equal. By using distance formula, we get

P1P32
2 = [Cos x – Cos (– y)]2 + [Sin x – Sin(–y]2
= (Cos x – Cos y)2 + (Sin x + Sin y)2
= Cos2x + Cos2y – 2 Cos x Cos y + Sin2x + Sin2y + 2Sin x Sin y
= 2 – 2 (Cos x Cos y – Sin x Sin y)

Also, P2P42

2 = [1 – Cos (x + y)]2 + [0 – Sin (x + y)]2
= 1 – 2Cos (x + y) + Cos2(x + y) + Sin2(x + y)
= 2 – 2 Cos (x + y)

Since P1P3 = P2P4, we have P1P32 = P2P42


Therefore, 2 –2 (Cos x Cos y – Sin x Sin y) = 2 – 2 Cos (x + y).


So, from the above discussion we get:

Cos (x + y) = Cos x Cos y – Sin x Sin y – (I)



Cosine: Sum and Difference of Angles


Sum of Angles: Cos (x + y)

From (I) we get:
Cos (x + y) = Cos x Cos y – Sin x Sin y


Difference of Angles: Cos (x – y)

Replacing y by (-y) in (I), we get:
Cos (x + y) = Cos x Cos y – Sin x Sin y
Cos (x + (-y)) = Cos x Cos (-y) – Sin x Sin (-y)
Cos (x – y) = Cos x Cos y + Sin x Sin y (From 1 and 2)



Sine: Sum and Difference of Angles

We will use Cos (Π2 – x) = Sin x to find out sum and difference of angles


Sum of Angles: Sin (x + y)

Sin (x + y) = Cos (Π2 – (x + y))
Sin (x + y) = Cos ((Π2 – x) – y)
= Cos (Π2 – x) Cos y + Sin (Π2 – x) Sin y (Using (I))
= Sin x Cos y + Cos x Sin y


Difference of Angles: Sin (x – y)

Replacing y by (-y) in sum of angles
Sin (x + (-y)) = Cos (Π2 – (x + (-y)))
Sin (x - y) = Cos ((Π2 – x) + y)
= Cos (Π2 – x) Cos y - Sin (Π2 – x) Sin y (Using (I))
= Sin x Cos y - Cos x Sin y



Tangent: Sum and Difference of Angles

We consider that x, y are not an odd multiple of  Π2


Sum of Angles: Tan (x + y)

Tan (x + y) = Sin (x+y)Cos (x+y)
Tan (x + y) = Sin x Cos y+Cos x Siny Cos x Cos y-Sin x Sin y

Dividing numerator and denominator by Cos x Cos y, we have:

Tan (x + y) = Tan x +Tan y1-Tan x Tan y


Difference of Angles: Tan (x – y)

Replacing y by (-y) in sum of angles
Tan (x – y) = Tan x- Tan y1+Tan x Tan y



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