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

Here we are going to see some practice questions to find the value of cos 2A when other trigonometric ratios given.

**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 = cos^{2}A - sin^{2}A

cos 2A = 1 - 2 sin^{2}A

cos 2A = 2cos^{2}A - 1

In the above formula, let us choose the 3^{rd} formula

cos 2A = 2cos^{2}A - 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 - 2sin^{2}A

= 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 - tan^{2}A) / (1 + tan^{2}A)

= (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

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

= (1/2)[cos^{2 }θ/2 + sin^{2} θ/2 - 2 sin^{ }θ/2 cos^{ }θ/2]

= (1/2) [1 - sin θ]

= (1/2) [1 - (1/25)]

= (1/2) (24/25)

sin^{2} (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

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

= (1/2)[cos^{2 }θ/2 + sin^{2} θ/2 - 2 sin^{ }θ/2 cos^{ }θ/2]

= (1/2) [1 - sin θ]

= (1/2) [1 - (8/9)]

= (1/2) (1/9)

cos^{2} (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, "Find the Value of Cos 2A When Other Trigonometric Ratios Given"

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