Answer :

I can help you find the value of tan(a + b) given sin(a) = 0.066 and cos(B) = 0.212, where both angles a and B are in Quadrant I.

Here's how we can approach this problem:

Tangent Addition Formula: We can't directly find tan(a + b) using the sine and cosine values of angles a and B. However, we can use the tangent addition formula:

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b))

Find tan(a) and tan(B):

We are given sin(a) = 0.066. Since a is in Quadrant I, we can use the arctangent function (tan^-1 or atan) to find tan(a). However, it's important to consider the quadrant to get the correct answer. You might need to use a calculator with an inverse tangent function that accounts for quadrants.

tan(a) = atan(sin(a)) = atan(0.066) ≈ 0.07 (rounded to two decimal places)

We are given cos(B) = 0.212. Since B is in Quadrant I, cos(B) is positive. We can use the arccosine function (cos^-1 or acos) to find tan(B). Again, consider the quadrant for accurate results.

Important Note: The arccosine function only outputs values in the range [0, pi]. Since tangent is positive in Quadrant I and negative in Quadrant III, and cosine is positive in both quadrants, we can find the tangent using the following relationship:

tan(B) = sqrt(1 - cos^2(B)) = sqrt(1 - (0.212)^2) ≈ 0.976 (rounded to two decimal places)

Apply the tangent addition formula:

Now that we have tan(a) and tan(B), we can plug them into the tangent addition formula:

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b))

tan(a + b) = (0.07 + 0.976) / (1 - 0.07 * 0.976)

tan(a + b) ≈ 1.045 / 0.931 (round after calculation)

tan(a + b) ≈ 1.123 (rounded to two decimal places)

Therefore, the value of tan(a + b) is approximately 1.12.