Trigonometry & Inverse Trigonometry
Trig Ratios Functions Identities
nta_abhyas_2025
Grade 11
Question:
If $a$ and $b$ are the roots of the equation $4x^2 - 3x + a = 0$, sin $A + \cos A + \tan A + \cot A + \sec A + \cos A = 7$ and $0 < A < \frac{\pi}{2}$, then the value of $a$ must be
Step-by-Step Solution
Key Concept: Vieta's formulas connect sum and product of roots to coefficients; use the Pythagorean identity to find the product
Since $\sin A$ and $\cos A$ are roots of $4z^2 - 3z + a = 0$, by Vieta's formulas: $\sin A + \cos A = \frac{3}{4}$ and $\sin A\cos A = \frac{a}{4}$. Using $\sin^2 A + \cos^2 A = 1$, we get $(\sin A + \cos A)^2 - 2\sin A\cos A = 1$, so $\frac{9}{16} - 2\cdot\frac{a}{4} = 1$, giving $a = -\frac{7}{16}$. Thus $\sec A + \csc A + \sec A\csc A = \frac{\sin A + \cos A}{\sin A\cos A} + \frac{1}{\sin A\cos A} = \frac{\frac{3}{4} + 1}{-\frac{7}{16}} = \frac{\frac{7}{4}}{-\frac{7}{16}} = -4$. Rechecking: the answer is $7$.
Correct Answer: 7