# FIND THE VALUE OF COS 2A WHEN OTHER TRIGONOMETRIC RATIOS GIVEN

Find the Value of  Cos 2A When Other Trigonometric Ratios Given :

In t his section, you will learn how to find the value of the trigonometric ratio cos 2A when other ratios are given.

## Finding the Value of Cos 2A From Other Trigonometric Ratios - Practice Questions

Question 1 :

Find the value of cos 2A, A lies in the first quadrant, when

(i) cos A = 15/17

Solution :

We have three formulas for cos 2A,

cos 2A  =  cos2A - sin2A

cos 2A  =  1 - 2 sin2A

cos 2A  =  2cos2A - 1

In the above formula, let us choose the 3rd formula

cos 2A  =  2cos2A - 1

=  2(15/17)2 - 1

=  2(225/289) - 1

=  (450 - 289)/289

cos 2A  =  161/289

(ii) sin A = 4/5

Solution :

cos 2A  =  1 - 2sin2A

=  1 - 2(4/5)2

=  1 - 2(16/25)

=  1 - (32/25)

=  (25 - 32)/25

cos 2A  =  -7/25

(iii) tan A = 16/63

Solution :

cos 2A  =  (1 - tan2A) / (1 + tan2A)

=  (1 - (16/63)2) / (1 + (16/63)2)

=  (1 - (256/3969)) / (1 + (256/3969))

=  (3713/3969) / (4225/3969)

=  3713/4225

Question 2 :

If θ is an acute angle, then find

(i) sin (π/4 - θ/2), when sin θ = 1/25

Solution :

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

sin (π/4 - θ/2)  =  sin π/4 cos θ/2 - cos π/4 sin θ/2

=  (1/√2)cos θ/2 - (1/√2) sin θ/2

sin (π/4 - θ/2)  =  (1/√2)[cos θ/2 - sin θ/2]

Taking squares on both sides

sin2 (π/4 - θ/2)  =  (1/√2)2[cos θ/2 - sin θ/2]2

=  (1/2)[cosθ/2 + sin2 θ/2 - 2 sin θ/2 cos θ/2]

=  (1/2) [1 - sin θ]

=  (1/2) [1 - (1/25)]

=  (1/2) (24/25)

sin2 (π/4 - θ/2)  =  12/25

sin (π/4 - θ/2)  =  12/25  =  2√3/5

(ii) cos (π/4 + θ/2), when sin θ = 8/9

Solution :

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

cos (π/4 + θ/2)  =  cos π/4 cos θ/2 - sin π/4 sin θ/2

=  (1/√2)cos θ/2 - (1/√2) sin θ/2

cos (π/4 + θ/2)  =  (1/√2)[cos θ/2 - sin θ/2]

Taking squares on both sides

cos2 (π/4 - θ/2)  =  (1/√2)2[cos θ/2 - sin θ/2]2

=  (1/2)[cosθ/2 + sin2 θ/2 - 2 sin θ/2 cos θ/2]

=  (1/2) [1 - sin θ]

=  (1/2) [1 - (8/9)]

=  (1/2) (1/9)

cos2 (π/4 + θ/2)  =  1/18

cos(π/4 + θ/2)  =  1/√18  =  1/3√2 After having gone through the stuff given above, we hope that the students would have understood how to find the value of the trigonometric ratio Cos 2A when other ratios are given.

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