Question

If \(2\cos\theta+3\sin\theta=3\) and \(\tan\theta\) is defined, then \(\tan\theta=\)

• A. \(\dfrac{5}{12}\)
• B. \(-\dfrac{5}{12}\)
• C. \(\dfrac{12}{5}\)
• D. \(-\dfrac{12}{5}\)

Hint:

Whenever expressions of the form \(a\cos\theta+b\sin\theta=c\) are given, combine them with \(\sin^2\theta+\cos^2\theta=1\) to determine the exact trigonometric ratios.

Answer 1

The Correct Option is A

Concept: Whenever an equation involving \(\sin\theta\) and \(\cos\theta\) is given, we use the fundamental identity \[ \sin^2\theta+\cos^2\theta=1. \] The given linear relation can be combined with this identity to determine the exact values of \(\sin\theta\) and \(\cos\theta\), and hence evaluate \(\tan\theta\).

Step 1: Square the given relation. Given, \[ 2\cos\theta+3\sin\theta=3. \] Squaring both sides, \[ 4\cos^2\theta+9\sin^2\theta+12\sin\theta\cos\theta=9. \] Using \[ \cos^2\theta=1-\sin^2\theta, \] we get \[ 4(1-\sin^2\theta)+9\sin^2\theta+12\sin\theta\cos\theta=9. \] Therefore, \[ 5\sin^2\theta+12\sin\theta\cos\theta=5. \]

Step 2: Use the given equation again. From \[ 2\cos\theta+3\sin\theta=3, \] we have \[ 2\cos\theta=3(1-\sin\theta). \] Thus, \[ \cos\theta=\frac{3(1-\sin\theta)}{2}. \] Substituting into \[ \sin^2\theta+\cos^2\theta=1, \] \[ \sin^2\theta+\frac{9(1-\sin\theta)^2}{4}=1. \] Multiplying by \(4\), \[ 4\sin^2\theta+9(1-2\sin\theta+\sin^2\theta)=4. \] \[ 13\sin^2\theta-18\sin\theta+5=0. \] Factorizing, \[ (13\sin\theta-5)(\sin\theta-1)=0. \] Hence, \[ \sin\theta=\frac{5}{13} \quad\text{or}\quad \sin\theta=1. \] Since \(\tan\theta\) is defined, \(\sin\theta=1\) is not possible because then \(\cos\theta=0\). Therefore, \[ \sin\theta=\frac{5}{13}. \]

Step 3: Find \(\cos\theta\). Using the given equation, \[ 2\cos\theta+3\left(\frac{5}{13}\right)=3. \] \[ 2\cos\theta=\frac{24}{13}. \] \[ \cos\theta=\frac{12}{13}. \]

Step 4: Calculate \(\tan\theta\). \[ \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}. \] Conclusion: \[ \boxed{\tan\theta=\frac{5}{12}} \]