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

If sin (x+y)=loge(x+y),thendydx is equal to