How do You Find the Values of sin2θ and cos2θ, when cosθ=12/13.

To find the trigonometric values of Sin 2θ and Cos 2θ we can make use of trigonometric ratios and formulas.

Answer: The Values of sin2θ and cos2θ, when cosθ=12/13 is Sin2θ =120/169 and Cos2θ = 119/169

Let us see the detailed solution,

Explanation: 

Given that,

Cosθ=12/13

Using the trignometric ratio for Cosθ, [ Cosθ = Adjacent/Hypotenuse  ]

==> Adjacent/Hypotenuse  = 12/13 

 Applying Pythagoras theorem, we can find the opposite side. 

 In a right angled triangle  if  Adjacent=12, Hypotenuse=13

Hypotenuse² = Adjacent ² + Opposite ²

Let the unknown opposite side be x.

Therefore,

132 = 122 + x²,

x² = 169-144

x² = 25

x = 5

Now, substituting these values into various trigonometric ratios , we get

Sinθ= Opposite / Hypotenuse

= 5/13

Cosθ = Adjacent / Hypotenuse

= 12/13

Now,

Using the trigonometric double angle identities 

Sin2θ  = 2SinθCosθ and  Cos2θ = Cos2θ - Sin2θ

Substituting the values of Sinθ and Cosθ in the above identities we get

Sin2θ  = 2SinθCosθ 

= 2×(5/13)×(12/13) 

= 120/169 

Cos2θ = Cos²θ-sin²θ

=(12/13)²- (5/13)²

= 119/169

Thus , The Values of sin2θ and cos2θ, when cosθ=12/13  is Sin2θ =  120/169 and  Cos2θ = 119/169