If $\tan\theta = \frac{3}{4}$, $(0^\circ | Mathematics MCQ | Examoogle

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If $\tan\theta = \frac{3}{4}$ , $(0°< \theta < 90°)$ , then the value of $\frac{4\sin\theta - \cos\theta}{4\sin\theta + \cos\theta}$ is:

1/2

3/4

2/3

1/4

Correct Answer

📐 MA • Math Direct SubstitutionMathematicsTrigonometry

Option A

1/2

Quick Summary:

Divide the numerator and denominator by $\cos \theta$ to directly obtain the expression in terms of $\tan \theta$ . Then, substitute $\tan \theta = 3/4$ .

📐MAMath SolutionDirect Substitution

📋 Given

Given $\tan \theta = \frac{3}{4}$ and $0°< \theta < 90°$ .

🔢 Formula Used

4sin⁡θ−cos⁡θ4sin⁡θ+cos⁡θ=4sin⁡θcos⁡θ−14sin⁡θcos⁡θ+1\frac{4\sin \theta - \cos \theta}{4\sin \theta + \cos \theta} = \frac{\frac{4\sin \theta}{\cos \theta} - 1}{\frac{4\sin \theta}{\cos \theta} + 1}4sinθ+cosθ4sinθ−cosθ​=cosθ4sinθ​+1cosθ4sinθ​−1​

⚡ Exam Hall Shortcut / Speed Trick

Divide the numerator and denominator by $\cos \theta$ to directly obtain the expression in terms of $\tan \theta$ . Then, substitute $\tan \theta = 3/4$ .

⚠️ Common Student Trap / Pitfall

Many students waste time by calculating $\sin \theta = 3/5$ and $\cos \theta = 4/5$ using a triangle, which is correct but slower than dividing by $\cos \theta$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Divide by $\cos \theta$

Divide both the numerator and the denominator by $\cos \theta$ to express the fraction in terms of $\tan \theta$ .

4sin⁡θcos⁡θ−14sin⁡θcos⁡θ+1\frac{4\frac{\sin \theta}{\cos \theta} - 1}{4\frac{\sin \theta}{\cos \theta} + 1}4cosθsinθ​+14cosθsinθ​−1​

Substitute $\tan \theta$

Since $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3}{4}$ , substitute this value into the expression.

4(34)−14(34)+1\frac{4(\frac{3}{4}) - 1}{4(\frac{3}{4}) + 1}4(43​)+14(43​)−1​

Simplify

Perform the arithmetic calculation in the numerator and denominator.

3−13+1=24=12\frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2}3+13−1​=42​=21​

✅

A is correct because the substitution results in the value 1/2.

Core Concepts Used

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Trigonometric Ratios Algebraic Substitution Tangent Identity

💡 EXAM TIP

This method of dividing by $\cos \theta$ is highly effective for any expression of the form $\frac{a \sin \theta + b \cos \theta}{c \sin \theta + d \cos \theta}$ found in Algebra and Calculus problems.

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