If $2\sin \theta + 15 \cos^2 \theta = 7, 0^\circ < \theta < 90^\circ$ then what is the value of $\frac{3 - \tan \theta}{2 + \tan \theta}$?
Answer & explanation
Correct answer: option 2
2 sinθ + 15 cos²θ = 7
{ using , sin²θ + cos²θ = 1 }
2 sinθ + 15( 1 - sin²θ ) = 7
15 sin²θ - 2sinθ - 8 = 0
15 sin²θ - 12sinθ + 10sinθ - 8 = 0
3 sinθ (5 sinθ - 4 ) + 2 (5 sinθ - 4 )= 0
Either (3 sinθ + 2 )= 0 OR (5 sinθ - 4 )= 0
sinθ = - \(\frac{2}{3}\) { not possible }
So , sinθ = \(\frac{4}{5}\)
By using pythagoras theorem ,
P² + B² = H²
4² + B² = 5²
B = 3
Now,
\(\frac{3 - tanθ}{2 + tanθ}\)
= \(\frac{3 - 4/3}{2 +4/3}\)
= \(\frac{1}{2}\)