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Question

Accepted Answer

Given tan A = 5/7.

Step 1: Express sin A and cos A in terms of tan A.
We know tan A = sin A / cos A = 5/7.

Let cos A = x, then sin A = (5/7) x.

Step 2: Use the Pythagorean identity sin^2 A + cos^2 A = 1.

(5/7 x)^2 + x^2 = 1
=> (25/49) x^2 + x^2 = 1
=> (25/49 + 1) x^2 = 1
=> (25/49 + 49/49) x^2 = 1
=> (74/49) x^2 = 1
=> x^2 = 49/74
=> x = sqrt(49/74) = 7 / sqrt(74)

Therefore, cos A = 7 / sqrt(74)
sin A = (5/7) * (7 / sqrt(74)) = 5 / sqrt(74)

Step 3: Calculate numerator and denominator of the given expression.

Numerator = 7 sin A - 3 cos A = 7 * (5 / sqrt(74)) - 3 * (7 / sqrt(74)) = (35 / sqrt(74)) - (21 / sqrt(74)) = 14 / sqrt(74)

Denominator = 7 sin A + 3 cos A = 7 * (5 / sqrt(74)) + 3 * (7 / sqrt(74)) = (35 / sqrt(74)) + (21 / sqrt(74)) = 56 / sqrt(74)

Step 4: Compute the fraction.

(7 sin A - 3 cos A) / (7 sin A + 3 cos A) = (14 / sqrt(74)) / (56 / sqrt(74)) = 14 / 56 = 1 / 4

Step 5: Add 4 to the fraction.

1/4 + 4 = 4 + 1/4 = 4 1/4

Hence, the value is 4 1/4.

Answer

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