Question

If

\[ \sin \theta + 2 \cos \theta = 1 \]

and

\[ \theta \text{ lies in the 4\textsuperscript{th} quadrant (not on coordinate axes), then } 7 \cos \theta + 6 \sin \theta =\ ? \]

• A. \( \frac{4}{17} \)
• B. \( 2 \)
• C. \( \frac{7}{17} \)
• D. \( \frac{4}{5} \)

Hint:

Use substitution and identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to simplify linear trigonometric equations. Always consider the quadrant for correct sign selection.

Answer 1

The Correct Option is B

Given: \[ \sin \theta + 2 \cos \theta = 1 \] Let’s solve by expressing this as a linear combination. Step 1: Assume \[ \sin \theta = a,
\cos \theta = b \] Then: \[ a + 2b = 1
\text{(1)} \] \[ a^2 + b^2 = 1
\text{(2)}
\text{(Pythagorean identity)} \] Step 2: From (1), express \( a = 1 - 2b \), and substitute into (2): \[ (1 - 2b)^2 + b^2 = 1 \Rightarrow 1 - 4b + 4b^2 + b^2 = 1 \Rightarrow 5b^2 - 4b = 0 \Rightarrow b(5b - 4) = 0 \Rightarrow b = 0 \text{ or } b = \frac{4}{5} \] Step 3: Since \( \theta \) lies in the 4th quadrant: - \( \cos \theta>0 \) - \( \sin \theta<0 \) So, choose \( \cos \theta = \frac{4}{5} \) Then from (1): \[ \sin \theta = 1 - 2 . \frac{4}{5} = 1 - \frac{8}{5} = -\frac{3}{5} \] Step 4: Evaluate the expression: \[ 7 \cos \theta + 6 \sin \theta = 7 . \frac{4}{5} + 6 . \left(-\frac{3}{5}\right) = \frac{28}{5} - \frac{18}{5} = \frac{10}{5} = \boxed{2} \]