What you will learn in this section
- Finding Cosne function formula for sum and addition of angles
- Finding Sine function formula for sum and addition of angles
- Finding Tangent function formula for sum and addition of angles
- Finding exact value of Sine, Cosine and Tangent value using these formula
What we assume you have already learnt
- Meaning and formulae of trigonometric expressions
- cos2A + sin2A = 1
- Cos(-A) = CosA and Sin(-A) = -SinA
Sum and difference of angles
If A and B are two angles,
- Sum of angles= A + B
- Difference of angle = A – B
- Multiple angles = nA where n is an integer. e.g. 2, 3A etc
When we say sum and difference of angles, we mean
Cosine of sum and difference of angles
Let us say we have a unit circle centered at O. We will take horizontal radius as base of our calculation. The point P(1,0) is the point where our reference/base meets the circumference.
Let A and B be points on circumference such that ∠ AOP = α and ∠ BOA = β;
This also means:
∠BOP = (α + β)
Let us draw an angle -β on OP such that it meets the circumference at B’. ∠ POB’ = -β
Coordinates of all points in the diagram:
- O(0,0)
- P(1,0)
- A (cosα, sinα)
- B (cos(α + β), sin(α + β))
- B'(cos(-β), sin(-β))
The distance between B and P is same as the distance between A and B’. i.e.
√(cos(α + β)-1)2 + (sin(α + β) – 0)2 = √(cosα-cos(-β))2 + (sinα – sin(-β))2
squaring both the sides:
(cos(α + β)-1)2 + (sin(α + β) – 0)2 = (cosα-cos(-β))2 + (sinα – sin(-β))2
or
cos2(α + β) -2cos(α + β) +1 + sin2(α + β) = (cosα-cos(β))2 + (sinα + sin(β))2
{since cos(-β) = cosβ and sin(-β) = -sinβ);
or
2 – 2cos(α + β) = cos2α – 2cosαcosβ + cos2β+ sin2α + 2sinαsinβ + sin2β
or
2 – 2cos(α + β) = <U>cos2α + sin2α </U>+ ??<U>cos2β + sin2β </U>+ 2sinαsinβ – 2cosαcosβ
or
2 – 2cos(α + β) = 2 + 2sinAsinB – 2cosαcosβ
or
-2cos(α + β) = 2sinAsinB – 2cosαcosβ
or
cos(α + β) = 2cosαcosβ – 2sinAsinB
this can be rewritten as:
cos(A + B) = cosAcosB – sinAsinB
In the above formula, if we replace B by -B, we will have:
cos(A + (- B)) = cosAcos(-B) – sinAsin(-B)
or
cos(A – B) = cosAcosB + sinAsinB [since cos(-B) = cos B and sin(-B) = -sinB
Usage
Find the value of cos75o
We can write cos75o as
cos(45 + 30)o = cos45ocos30o – sin45osin30o
= (1/√2)*(√3/2) – (1/√2)*(1/2)
= (√3-1)/2√2
or
Sine of sum and difference of angles
In the figure given below, Let us say ∠ABC = α and ∠PBA = β ∠PBC = α+β Also, BP = CA AP is parallel to BC
Let us extend BP and CA. Draw a perpendicular on AB at A such that it meets BP at O. Also draw OR parallel to BC. So, ∠OAR = α
Let’s say AB = h, OB = T and OA = x
then sinβ = x/T or x = T*sinβ and h = T*cosβ
AC = hsinα = T*sinα*cosβ —– (i)
AR = xcosα = T*cosα*sinβ —– (ii)
sin(α+β) = RC/T or
sin(α+β) = (AR + AC)/T —– (iii)
substituting values of AR and AC from (i) and (ii) into (iii), we get
sin(α+β) = (T*sinα*cosβ + T*cosα*sinβ)/T
dividing numerator and denominator by T, we get
For multiple and submultiple angles, please click here.