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

The correct option is: (C) 16x² - 12x + 1 = 0.

(A) The sum of the roots can be expressed as the combination of trigonometric values:

sin⁡2(18∘)+cos⁡2(36∘)=(45−15)2+(45+15)2sin2(18∘)+cos2(36∘)=(54​−51​)2+(54​+51​)2

=1625+1625=3225=2516​+2516​=2532​

=1625[(5+1)2+(5−1)2]=2516​[(5+1)2+(5−1)2]

=1625⋅2⋅(5+1)=2516​⋅2⋅(5+1)

=3225⋅6=2532​⋅6

=19225=25192​

Hence, the sum of the roots is 1922525192​.

The product of the roots can be expressed similarly:

sin⁡2(18∘)⋅cos⁡2(36∘)=(45−15)2⋅(45+15)2sin2(18∘)⋅cos2(36∘)=(54​−51​)2⋅(54​+51​)2

=1625⋅1625=256625=2516​⋅2516​=625256​

Therefore, the quadratic equation can be written as:

cos⁡2(36∘))x2−(sin2(18∘)+cos2(36∘))x+(sin2(18∘)⋅cos2(36∘))

625x2−25192​x+625256​

Finally, we can multiply the entire equation by 2525 to get rid of the fractions:

2525x2−192x+25256​

Multiplying by 2525 gives:

16x2−192x+256=0(A) The sum of the roots can be expressed as the combination of trigonometric values:

sin⁡2(18∘)+cos⁡2(36∘)=(45−15)2+(45+15)2sin2(18∘)+cos2(36∘)=(54​−51​)2+(54​+51​)2

=1625+1625=3225=2516​+2516​=2532​

=1625[(5+1)2+(5−1)2]=2516​[(5+1)2+(5−1)2]

=1625⋅2⋅(5+1)=2516​⋅2⋅(5+1)

=3225⋅6=2532​⋅6

=19225=25192​

Hence, the sum of the roots is 1922525192​.

The product of the roots can be expressed similarly:

sin⁡2(18∘)⋅cos⁡2(36∘)=(45−15)2⋅(45+15)2sin2(18∘)⋅cos2(36∘)=(54​−51​)2⋅(54​+51​)2

=1625⋅1625=256625=2516​⋅2516​=625256​

Therefore, the quadratic equation can be written as:

(sin2(18∘)+cos2(36∘))x+(sin2(18∘)⋅cos2(36∘))

625x2−25192​x+625256​

Finally, we can multiply the entire equation by 2525 to get rid of the fractions:

2525x2−192x+25256​

Multiplying by 2525 gives:

16x2−192x+256=0