COS A PLUS B FORMULA

cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B 

Express each of the following in the form sin α, where α is acute.

Problem 1 :

cos 14º cos 39º - sin 14º sin 39º

Solution :

cos (A + B) = cos A cos B - sin A sin B 

= cos 14º cos 39º - sin 14º sin 39º

= cos (14º + 39º)

= cos 53º

Problem 2 :

Express as a single trigonometric ratio.

a. cos A cos 2A - sin A sin 2A

b. cos A cos 3A + sin A sin 3A

Solution :

a. 

cos (A + B) = cos A cos B - sin A sin B 

= cos A cos 2A - sin A sin 2A

= cos (A + 2A)

=  cos 3A

b.

cos (A - B) = cos A cos B + sin A sin B 

= cos A cos 3A + sin A sin 3A

= cos (A + 3A)

=  cos 4A

Problem 3 :

cos 38º cos 8º + sin 38º sin 8º is equal to.

a)  cos 30º  b)  cos 60º   c)   cos 45º   d)  cos 38º

Solution  :

cos (A - B) = cos A cos B + sin A sin B 

= cos 38º cos 8º + sin 38º sin 8º

= cos (38º - 8º)

= cos 30º

So, option a) is correct.

Problem 4 :

Find the value of k such that for all real values of x.

cos x + 𝜋3 - cos x - 𝜋3 k sin x

Solution :

Given, cos x + 𝜋3 - cos x - 𝜋3 k sin xcos c - cos d = -2 sin c + d2· sin c - d2 -2 sin x + 𝜋3 + x - 𝜋32 · sin x + 𝜋3 - x - 𝜋32 = k sin x -2 sin x + 𝜋3 + x - 𝜋32 · sin x + 𝜋3 - x + 𝜋32 = k sin x -2 sin 2x 2 · sin 2𝜋32 = k sin x -2 sin 2x 2 · sin 2𝜋3 × 12 = k sin x-2 sin x · sin 2𝜋6 = k sin x-2 sin x · sin 𝜋3 = k sin x-2 sin x ·32= k sin x- 3 sin x = k sin xk = - 3 sin xsin xk = -3

So, the value of k is -√3.

Problem 5 :

Prove each identity.

cos x - cos x - 𝜋3 cos x + 𝜋3

Solution :

Given, cos x - cos x - 𝜋3 cos x + 𝜋3cos x = cos x + 𝜋3 + cos x - 𝜋3cos (a + b) = cos a cos b - sin a sin bcos (a - b) = cos a cos b + sin a sin bcos x = cos x cos 𝜋3 - sin x sin 𝜋3 +cos x cos 𝜋3 + sin x sin 𝜋3cos x = 2 cos x cos 𝜋3 cos x = 2 cos x 12cos x = cos x

Problem 6 :

Rewrite each expression as a single trigonometric ratio.

cos 4x cos 3x - sin 4x sin 3x

Solution :

cos (A + B) = cos A cos B - sin A sin B 

= cos 4x cos 3x - sin 4x sin 3x

= cos (4x + 3x)

= cos 7x

Problem 7 :

Rewrite each expression as a single trigonometric ratio, and then evaluate the ratio.

cos 5𝜋12 cos 𝜋12 + sin 5𝜋12 sin 𝜋12

Solution :

cos (A - B) = cos A cos B + sin A sin B= cos 5𝜋12 cos 𝜋12 + sin 5𝜋12 sin 𝜋12= cos 5𝜋12 - 𝜋12=cos 4𝜋12 = cos 𝜋3= 12

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