Question

The quadratic equation whose roots are sin²18° and cos² 36° is

• A. 16x² - 12x - 1 = 0
• B. 16x² - 12x + 4 = 0
• C. 16x² - 12x + 1 = 0
• D. 16x² + 12x + 1 = 0

Answer 1

The Correct Option is C

Step 1: Understanding the roots
The roots of the quadratic equation are \( \sin^2 18^\circ \) and \( \cos^2 36^\circ \). We need to form the quadratic equation with these as roots. The general form of the quadratic equation with roots \( r_1 \) and \( r_2 \) is given by:
\( (x - r_1)(x - r_2) = 0 \), where \( r_1 = \sin^2 18^\circ \) and \( r_2 = \cos^2 36^\circ \).

Step 2: Using the identity for a quadratic equation
The sum of the roots of the quadratic equation is given by \( r_1 + r_2 = \sin^2 18^\circ + \cos^2 36^\circ \), and the product of the roots is \( r_1 \times r_2 = \sin^2 18^\circ \times \cos^2 36^\circ \).

Step 3: Calculate sum and product of the roots
First, we need the values of \( \sin 18^\circ \) and \( \cos 36^\circ \). Using a calculator or trigonometric tables, we find:
\( \sin 18^\circ \approx 0.3090 \) and \( \cos 36^\circ \approx 0.8090 \).
Now, calculate their squares:
\( \sin^2 18^\circ \approx (0.3090)^2 = 0.0955 \) and \( \cos^2 36^\circ \approx (0.8090)^2 = 0.6545 \).
Thus, the sum of the roots is:
\( r_1 + r_2 = 0.0955 + 0.6545 = 0.75 \).
The product of the roots is:
\( r_1 \times r_2 = 0.0955 \times 0.6545 = 0.0625 \).

Step 4: Form the quadratic equation
The quadratic equation with roots \( r_1 \) and \( r_2 \) is:
\( x^2 - (r_1 + r_2)x + r_1 \times r_2 = 0 \). Substituting the values we found:
\( x^2 - 0.75x + 0.0625 = 0 \).

Step 5: Compare with the options
We now compare the derived equation \( x^2 - 0.75x + 0.0625 = 0 \) with the given options. The equation closest to this form is:
\( 16x^2 - 12x + 1 = 0 \).

Final Answer:
The quadratic equation whose roots are \( \sin^2 18^\circ \) and \( \cos^2 36^\circ \) is:
\( 16x^2 - 12x + 1 = 0 \).