Question

If \( 7\cos\theta - \sin\theta = 5 \) and \( \tan\theta>0 \), then \( \tan\theta = \)

• A. \( \dfrac{7}{12} \)
• B. \( \dfrac{3}{4} \)
• C. \( \dfrac{4}{3} \)
• D. \( \dfrac{12}{7} \)

Hint:

When dealing with expressions involving trigonometric identities and conditions like \( \tan\theta>0 \), always use substitution and Pythagorean identities for a system of equations.

Answer 1

The Correct Option is B

Step 1: Let \( \cos\theta = x \) and \( \sin\theta = y \).
Then from the identity \( x^2 + y^2 = 1 \).
Also given:
\[ 7x - y = 5 \tag{1} \] Step 2: From identity:
\[ x^2 + y^2 = 1 \tag{2} \] Step 3: Solve equation (1) for \( y \):
\[ y = 7x - 5 \] Substitute into (2):
\[ x^2 + (7x - 5)^2 = 1 \] \[ x^2 + 49x^2 - 70x + 25 = 1 \Rightarrow 50x^2 - 70x + 24 = 0 \] Step 4: Solve the quadratic:
\[ x = \dfrac{70 \pm \sqrt{(-70)^2 - 4 \cdot 50 \cdot 24}}{2 \cdot 50} = \dfrac{70 \pm \sqrt{4900 - 4800}}{100} = \dfrac{70 \pm 10}{100} \Rightarrow x = \dfrac{80}{100} = \dfrac{4}{5}, \quad \text{or} \quad \dfrac{60}{100} = \dfrac{3}{5} \] Step 5: Use both values of \( x \) to get corresponding \( y \):
If \( x = \dfrac{4}{5} \), then
\[ y = 7 \cdot \dfrac{4}{5} - 5 = \dfrac{28 - 25}{5} = \dfrac{3}{5} \Rightarrow \tan\theta = \dfrac{y}{x} = \dfrac{3}{4} \] If \( x = \dfrac{3}{5} \), then
\[ y = 7 \cdot \dfrac{3}{5} - 5 = \dfrac{21 - 25}{5} = -\dfrac{4}{5} \Rightarrow \tan\theta = \dfrac{-4}{3} \quad (\text{not allowed}) \] Only valid value: \( \tan\theta = \dfrac{3}{4} \)