If 7 tan θ = 3, and θ is an acute angle, then what is the value of (5 sin θ - cos θ) / (5 sin θ + 2 cos θ)?

Question

Accepted Answer

Given: 7 tan θ = 3, θ is acute.

Step 1: Find tan θ.
7 tan θ = 3 ⇒ tan θ = 3/7

Step 2: Express sin θ and cos θ in terms of tan θ.
Let sin θ = x, cos θ = y. Then tan θ = x/y = 3/7 ⇒ x : y = 3 : 7

Let sin θ = 3k, cos θ = 7k
But sin²θ + cos²θ = 1
(3k)² + (7k)² = 1
9k² + 49k² = 1
58k² = 1 ⇒ k² = 1/58 ⇒ k = 1/√58

So,
sin θ = 3/√58
cos θ = 7/√58

Step 3: Substitute in the expression:
(5 sin θ - cos θ) / (5 sin θ + 2 cos θ)
= [5*(3/√58) - (7/√58)] / [5*(3/√58) + 2*(7/√58)]
= [ (15 - 7)/√58 ] / [ (15 + 14)/√58 ]
= (8/√58) / (29/√58)
= 8/29

But this is not matching the correct answer. Let's check the calculation carefully.

Wait, let's recalculate:
Numerator: 5*sin θ - cos θ = 5*(3/√58) - (7/√58) = (15 - 7)/√58 = 8/√58
Denominator: 5*sin θ + 2*cos θ = 5*(3/√58) + 2*(7/√58) = (15 + 14)/√58 = 29/√58
So, (8/√58) / (29/√58) = 8/29

But the answer key says 7/29 is correct. Let's check the options and calculation again.

Given tan θ = 3/7, so sin θ = 3/√(3²+7²) = 3/√58, cos θ = 7/√58.

Numerator: 5*sin θ - cos θ = 5*(3/√58) - (7/√58) = (15 - 7)/√58 = 8/√58
Denominator: 5*sin θ + 2*cos θ = 5*(3/√58) + 2*(7/√58) = (15 + 14)/√58 = 29/√58
So, (8/√58) / (29/√58) = 8/29

So, the correct answer is 8/29, not 7/29. The answer key may be incorrect. The calculation is correct.

Final Answer: 8/29 (Option 1)

Step-by-step summary:
1. tan θ = 3/7
2. sin θ = 3/√58, cos θ = 7/√58
3. Substitute into the given expression and simplify:
   (5 sin θ - cos θ) / (5 sin θ + 2 cos θ) = 8/29

Answer

Answer

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