How can I determine the values of sin 2theta and cos 2theta if cos theta is 12/13?

To find the values of sin 2theta and cos 2theta when cos theta is given as 12/13, we can use the double angle formulas in trigonometry:

First, we need to determine the missing value: sin theta. Using the Pythagorean identity:

sin²θ + cos²θ = 1

We know that:

cos²θ = (12/13)² = 144/169

Now, substituting into the Pythagorean identity:

sin²θ + 144/169 = 1

Solving for sin²θ:

sin²θ = 1 - 144/169 = 25/169

Taking the square root of both sides, we find:

sinθ = ±√(25/169) = ±5/13

Since we typically consider the range of angles for θ where both sine and cosine are positive in the first quadrant, we take sin θ = 5/13.

Now we can calculate sin 2theta:

sin 2θ = 2 * sin θ * cos θ = 2 * (5/13) * (12/13) = 120/169

Next, let’s calculate cos 2theta:

cos 2θ = cos²θ - sin²θ = (12/13)² - (5/13)² = 144/169 - 25/169 = 119/169

In conclusion, the values are: