Trigonometric Addition Formulas

The trigonometric addition formulas can be applied to simplify a complicated expression or find an exact value when you are with only some trigonometric values. For instance, if you want the Sine of 15 degrees, you can use a subtraction formula to calculate sin(15) as sin(45-30).

FORMULAS

  • sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
  • sin(a-b)= sin(a)cos(b)-cos(a)sin(b)
  • cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
  • cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
  • \(\tan (a+b) = \frac{\tan a + \tan b}{1 – \tan a \tan b}\)
  • \(\tan (a+b) = \frac{\tan a – \tan b}{1 + \tan a \tan b}\)

Let’s solve some examples:

Example 1

Verify the identity

\(\cos (a+b). \cos (a-b) = \cos^{2}a-\sin^{2}b\)

Solution

cos(a+b) cos(a-b) = (cos(a)cos(b)-sin(a)sin(b)) (cos(a)cos(b) + sin(a)sin(b))

This gives,

\(\cos(a+b) \cos(a-b)= \cos^{2}a \cos^{2}b – \sin^{2}a \sin^{2}b\) …………(i)

But,

\(\sin^{2}a \sin^{2}b = (1-\cos^{2}a).(1- \cos^{2}b) = 1-\cos^{2}a-\cos^{2}b +\cos^{2}a\cos^{2}b\) ………(ii)

Equating the value of (ii) in (i), we have

\(\cos(a+b) \cos(a-b)= \cos^{2}a \cos^{2}b – 1 + \cos^{2}a + \cos^{2}b – \cos^{2}a\cos^{2}b\)

cos2(a)cos2(b) – (1 – cos2(a) – cos2(b) + cos2(a) cos2(b)) = cos2(a) + cos2(b) – 1

\(\Rightarrow \cos(a+b) \cos(a-b)= \cos^{2}a + \cos^{2}b – 1\)

We know, \(\cos^{2}b = 1 – \sin^{2}b\)

Therefore, we get

cos(a + b) cos(a-b) = cos2(a) – sin2(b)

Example 2

Calculate the exact value of \(\cos (165^{\circ})\).

Solution

By applying the addition formulas for the cosine function, we get

cos(165) = cos(120+45) = cos(120)cos(45) – sin(120)sin(45)

Since we know that,

cos(120) = -½

sin(120) = 3/2

cos(45) = 2/2

sin(45) = √2/2

Therefore, we get

cos(165) = \(-\frac{1}{2}.2\sqrt{2} – \frac{\sqrt{3}}{2}. \frac{\sqrt{2}}{2}\)

\(= -\frac{\sqrt{2}+\sqrt{6}}{4}\)<

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Practise This Question

In a group of 70 people, 37 like coffee, 52 like tea and each person like at least one of the two drinks. The number of persons liking both coffee and tea is: