If 7 tan θ = 3, then find the value of (7 sin θ + 5 cos θ) / (7 sin θ − 2 cos θ).

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Accepted Answer

Given: 7 tan θ = 3
=> tan θ = 3/7

Let sin θ = x and cos θ = y.
We know tan θ = sin θ / cos θ = x / y = 3/7
=> x = (3/7) y

Using the Pythagorean identity:
 sin² θ + cos² θ = 1
=> x² + y² = 1
=> ((3/7) y)² + y² = 1
=> (9/49) y² + y² = 1
=> (9/49) y² + (49/49) y² = 1
=> (58/49) y² = 1
=> y² = 49/58
=> y = sqrt(49/58) = 7 / sqrt(58)

Then x = (3/7) y = (3/7) * (7 / sqrt(58)) = 3 / sqrt(58)

Now calculate the expression:
(7 sin θ + 5 cos θ) / (7 sin θ − 2 cos θ)
= (7x + 5y) / (7x − 2y)
= (7 * (3 / sqrt(58)) + 5 * (7 / sqrt(58))) / (7 * (3 / sqrt(58)) − 2 * (7 / sqrt(58)))
= (21 / sqrt(58) + 35 / sqrt(58)) / (21 / sqrt(58) − 14 / sqrt(58))
= (56 / sqrt(58)) / (7 / sqrt(58))
= 56 / 7 = 8

Rechecking the calculation shows the value is 8, which is not among the options. Let's carefully re-examine the steps:

Step 1: tan θ = 3/7

Step 2: sin θ = (3/7) cos θ

Step 3: sin² θ + cos² θ = 1
=> ((3/7) cos θ)² + cos² θ = 1
=> (9/49) cos² θ + cos² θ = 1
=> (9/49 + 1) cos² θ = 1
=> (9/49 + 49/49) cos² θ = 1
=> (58/49) cos² θ = 1
=> cos² θ = 49/58
=> cos θ = 7 / sqrt(58)

Step 4: sin θ = (3/7) * (7 / sqrt(58)) = 3 / sqrt(58)

Step 5: Calculate numerator and denominator:
Numerator = 7 sin θ + 5 cos θ = 7 * (3 / sqrt(58)) + 5 * (7 / sqrt(58)) = (21 + 35) / sqrt(58) = 56 / sqrt(58)
Denominator = 7 sin θ − 2 cos θ = 7 * (3 / sqrt(58)) − 2 * (7 / sqrt(58)) = (21 − 14) / sqrt(58) = 7 / sqrt(58)

Step 6: Expression = (56 / sqrt(58)) / (7 / sqrt(58)) = 56 / 7 = 8

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