What is The Formula of Cos(a+b)?

Cos(A+B) Formula

Are you looking Cos(a+b) formula? Here you can get all information such as how to proof this formula as well as verify through values. First of all we are going to share the formula of Cos(a+b):

\(\cos(a + b) = \cos(a) \cos(b) – \sin(a) \sin(b)\)

Cos (A+B) Proof

From triangle ACE we get, ∠EAC = 90° – ∠ACE = ∠ECO = alternate ∠COX = α. Now, from the right-angled triangle AOB we get,

cos (α + β) = \(\frac{OB}{OA}\)

= \(\frac{OD – BD}{OA}\)

= \(\frac{OD}{OA} – \frac{BD}{OA}\)

= \(\frac{OD}{OA} – \frac{EC}{OA}\)

= \(\frac{OD}{OC} \times \frac{OC}{OA} – \frac{EC}{AC} \times \frac{AC}{OA}\)

= cos α cos β – sin ∠EAC sin β

= cos α cos β – sin α sin β, (since we know, ∠EAC = α)

Therefore, \(\cos(a + b) = \cos(a) \cos(b) – \sin(a) \sin(b)\) Proved

Cos (A+B) Verification

Need to verify cos(a+b)formula is right or wrong. put the value of a =45° degree and b=30° degree

put the value of a and b in the LHS

cos(a+b) = cos(45°+30°)

= cos(75°) = \(\frac{√3  –  1}{2√2}\)

put the value of a and b in the RHS

=> cos(a) cos(b) – sin(a) sin(b)

=> cos(45°) cos(30°) – sin(45°) sin(30°)

Put the Value

= \(\frac{1}{√2} \times \frac{√3}{2} – \frac{1}{√2} \times \frac{1}{2}\)

= \(\frac{√3  –  1}{2√2}\)

Therefore LHS = RHS [Note: LHS =Left hand Side, RHS =Right hand side]

Summary cos(a+b)

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