I don\'t follow your thinking

If the do, it is an easy problem.
draw a line from the vertex through the two centres.
Let the distance between the vertex and the centre of the smaller circle be x
then the distance from the vertex to the centre of the larger circle is x+6

sin (theta/2) = 2/x
sin (theta/2) = 4/(x+6)
then 2/x = 4/(x+6)
which solves for x=6

then sin (theta/2) = 2/6
theta/2 = 19.47º
theta = 38.942º or 38.9º to the nearest tenth.

circles do touch

what is this half angle identity thingy

sine theta/2 is equal to 2/x

why please explain

the circles did not touch in your diagram.

In your diagram, theta was the entire angle between the two langent lines.
So the line to the centre would bisect the angle, in other words, theta/2.

Can you not see that we now created right-angles triangles?
I then just used the primary trig relationship, sin(angle) = opposite/hypotenuse
thus, sin(theta/2) = 2/x etc

I apologize if my explanation was not clear. Let me break it down step by step to help you understand the reasoning behind the solution.

The problem involves two circles and a vertex (point of intersection) formed by drawing a line through the centers of the circles. To find the angle, we need to determine the value of theta.

1. Draw a line from the vertex through the two centers: This step helps us visualize and understand the problem better.

2. Let the distance between the vertex and the center of the smaller circle be x: By assigning a variable, x, to the distance, we can use it to form equations and solve for theta.

3. The distance from the vertex to the center of the larger circle is x + 6: Since the larger circle is 6 units away from the smaller circle's center, we can express its distance from the vertex as x + 6.

4. Use trigonometry to form equations: By considering the angles formed at the vertex, we can use trigonometric ratios to form equations.

- sin (theta/2) = 2/x: Here, we use the sine ratio, as the opposite side is 2 and the adjacent side is x.
- sin (theta/2) = 4/(x + 6): Again, we use the sine ratio, but this time the opposite side is 4 and the adjacent side is x + 6.

5. Set up an equation to solve for x: Equate the two equations formed in step 4 to find the value of x.

- 2/x = 4/(x + 6): By cross-multiplying, we can eliminate the fractions and obtain a new equation.

6. Solve for x: Simplify and solve the equation to find the value of x.

- Multiplying both sides of the equation by x and distributing, we get 2(x + 6) = 4x.
- Expanding, we have 2x + 12 = 4x.
- Subtracting 2x from both sides, we get 12 = 2x.
- Dividing both sides by 2, we find x = 6.

7. Substitute the value of x back into one of the original equations: We substitute x = 6 into either sin (theta/2) = 2/x or sin (theta/2) = 4/(x + 6).

- sin (theta/2) = 2/6: Simplifying, we have sin (theta/2) = 1/3.

8. Solve for theta: Double the angle obtained from the previous step to find the value of theta.

- theta/2 = arcsin(1/3): Taking the inverse sine of both sides to isolate theta/2.
- theta/2 ≈ 19.47º: Evaluating the arcsine of 1/3.
- theta ≈ 2 * 19.47º: Doubling the angle to find theta.
- theta ≈ 38.94º: Approximating theta to the nearest tenth.

Therefore, the angle theta is approximately 38.9º to the nearest tenth.