If sin a = 1/10 and cos b = 2/29, find tan(2a) where a and b lie between 0 to π/4.

To find tan(2a), we can use the double angle formula for tangent:

tan(2a) = 2tan(a) / (1 – tan²(a))

First, we need to find tan(a). We know:

sin(a) = 1/10

Now, we can calculate cos(a) using the Pythagorean identity:

cos(a) = √(1 – sin²(a))

Calculating that:

cos(a) = √(1 – (1/10)²) = √(1 – 1/100) = √(99/100) = √99/10

Now, we can find tan(a):

tan(a) = sin(a) / cos(a) = (1/10) / (√99/10) = 1 / √99

Now we can compute tan(2a):

tan(2a) = 2tan(a) / (1 – tan²(a))

Calculating tan²(a):

tan²(a) = (1 / √99)² = 1 / 99

Substituting back into the formula:

tan(2a) = 2(1 / √99) / (1 – 1/99) = 2/√99 / (98/99) = (2 * 99) / (√99 * 98)

Simplifying further gives:

tan(2a) = 198 / (√99 * 98)

So the final answer is:

tan(2a) = 198 / (√99 * 98)