If 3 sin A + 4 cos A = 5, then find the value of tan A.

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Given: 3 sin A + 4 cos A = 5

Step 1: Use the identity that for any angle A, a sin A + b cos A = R sin (A + α), where R = sqrt(a^2 + b^2) and α = arctan(b/a).

Here, a = 3, b = 4.

Calculate R:
R = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

So, 3 sin A + 4 cos A = 5 sin (A + α) = 5.

This implies sin (A + α) = 1.

Therefore, A + α = 90° (or π/2 radians).

Calculate α:
α = arctan(b/a) = arctan(4/3).

So, A = 90° - α = 90° - arctan(4/3).

Step 2: Find tan A:

Using the formula:

If α = arctan(4/3), then tan α = 4/3.

Then tan A = tan(90° - α) = cot α = 1 / tan α = 1 / (4/3) = 3/4.

But the options do not have 3/4 as correct answer; let's verify carefully.

Wait, the original equation is 3 sin A + 4 cos A = 5.

Since maximum value of 3 sin A + 4 cos A is sqrt(3^2 + 4^2) = 5, and the equation equals 5, this means sin (A + α) = 1.

So, A + α = 90°.

We have α = arctan(4/3).

Therefore, A = 90° - arctan(4/3).

Now, tan A = tan(90° - arctan(4/3)) = cot(arctan(4/3)) = 1 / (4/3) = 3/4.

Hence, tan A = 3/4.

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