To find the value of 3 cos 36 ∘ + 5 sin 18 ∘ 5 cos 36 ∘ − 3 sin 18 ∘ \frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}} 5 c o s 3 6 ∘ − 3 s i n 1 8 ∘ 3 c o s 3 6 ∘ + 5 s i n 1 8 ∘ in the form a 5 − b c \frac{a \sqrt{5}-b}{c} c a 5 − b , we need to simplify the given expression. Let's start by using some fundamental trigonometric identities.
We know that:
cos 36 ∘ = 5 + 1 4 \cos 36^{\circ} = \frac{\sqrt{5} + 1}{4} cos 3 6 ∘ = 4 5 + 1
sin 18 ∘ = 5 − 1 4 \sin 18^{\circ} = \frac{\sqrt{5} - 1}{4} sin 1 8 ∘ = 4 5 − 1
First, substitute these values into the expression:
3 ⋅ 5 + 1 4 + 5 ⋅ 5 − 1 4 5 ⋅ 5 + 1 4 − 3 ⋅ 5 − 1 4 \frac{3 \cdot \frac{\sqrt{5} + 1}{4} + 5 \cdot \frac{\sqrt{5} - 1}{4}}{5 \cdot \frac{\sqrt{5} + 1}{4} - 3 \cdot \frac{\sqrt{5} - 1}{4}} 5 ⋅ 4 5 + 1 − 3 ⋅ 4 5 − 1 3 ⋅ 4 5 + 1 + 5 ⋅ 4 5 − 1
Simplify the numerator and the denominator:
Numerator: 3 ⋅ 5 + 1 4 + 5 ⋅ 5 − 1 4 = 3 ( 5 + 1 ) + 5 ( 5 − 1 ) 4 = 3 5 + 3 + 5 5 − 5 4 = 8 5 − 2 4 = 2 5 − 1 2 3 \cdot \frac{\sqrt{5} + 1}{4} + 5 \cdot \frac{\sqrt{5} - 1}{4} = \frac{3(\sqrt{5} + 1) + 5(\sqrt{5} - 1)}{4} = \frac{3\sqrt{5} + 3 + 5\sqrt{5} - 5}{4} = \frac{8\sqrt{5} - 2}{4} = 2 \sqrt{5} - \frac{1}{2} 3 ⋅ 4 5 + 1 + 5 ⋅ 4 5 − 1 = 4 3 ( 5 + 1 ) + 5 ( 5 − 1 ) = 4 3 5 + 3 + 5 5 − 5 = 4 8 5 − 2 = 2 5 − 2 1
Denominator: 5 ⋅ 5 + 1 4 − 3 ⋅ 5 − 1 4 = 5 ( 5 + 1 ) − 3 ( 5 − 1 ) 4 = 5 5 + 5 − 3 5 + 3 4 = 2 5 + 8 4 = 5 + 4 2 5 \cdot \frac{\sqrt{5} + 1}{4} - 3 \cdot \frac{\sqrt{5} - 1}{4} = \frac{5(\sqrt{5} + 1) - 3(\sqrt{5} - 1)}{4} = \frac{5\sqrt{5} + 5 - 3\sqrt{5} + 3}{4} = \frac{2\sqrt{5} + 8}{4} = \frac{\sqrt{5}+4}{2} 5 ⋅ 4 5 + 1 − 3 ⋅ 4 5 − 1 = 4 5 ( 5 + 1 ) − 3 ( 5 − 1 ) = 4 5 5 + 5 − 3 5 + 3 = 4 2 5 + 8 = 2 5 + 4
Now combine the simplified numerator and denominator:
2 5 − 1 2 5 + 4 2 = ( 2 5 − 1 2 ) ⋅ 2 5 + 4 = 4 5 − 1 5 + 4 \frac{2 \sqrt{5} - \frac{1}{2}}{\frac{\sqrt{5}+4}{2}} = \frac{(2 \sqrt{5} - \frac{1}{2}) \cdot 2}{\sqrt{5}+4} = \frac{4 \sqrt{5} - 1}{\sqrt{5}+4} 2 5 + 4 2 5 − 2 1 = 5 + 4 ( 2 5 − 2 1 ) ⋅ 2 = 5 + 4 4 5 − 1
Rationalizing the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator:
( 4 5 − 1 ) ( 5 − 4 ) ( 5 + 4 ) ( 5 − 4 ) \frac{(4 \sqrt{5} - 1)(\sqrt{5}-4)}{(\sqrt{5}+4)(\sqrt{5}-4)} ( 5 + 4 ) ( 5 − 4 ) ( 4 5 − 1 ) ( 5 − 4 )
The denominator simplifies to:
5 2 − 4 2 = 5 − 16 = − 11 \sqrt{5}^2 - 4^2 = 5 - 16 = -11 5 2 − 4 2 = 5 − 16 = − 11
The numerator simplifies to:
( 4 5 − 1 ) ( 5 − 4 ) = ( 4 5 ⋅ 5 − 4 ⋅ 4 5 − 1 ⋅ 5 + 1 ⋅ 4 ) = ( 20 − 16 5 − 5 + 4 ) = 24 − 17 5 (4 \sqrt{5} - 1)(\sqrt{5}-4) = (4 \sqrt{5} \cdot \sqrt{5} - 4 \cdot 4 \sqrt{5} - 1 \cdot \sqrt{5} + 1 \cdot 4) = (20 - 16 \sqrt{5} - \sqrt{5} + 4) = 24 - 17 \sqrt{5} ( 4 5 − 1 ) ( 5 − 4 ) = ( 4 5 ⋅ 5 − 4 ⋅ 4 5 − 1 ⋅ 5 + 1 ⋅ 4 ) = ( 20 − 16 5 − 5 + 4 ) = 24 − 17 5
Combining them, we get:
24 − 17 5 − 11 = 17 5 − 24 11 \frac{24 - 17\sqrt{5}}{-11} = \frac{17\sqrt{5}-24}{11} − 11 24 − 17 5 = 11 17 5 − 24
Thus, a = 17 , b = 24 , c = 11 a = 17, b = 24, c = 11 a = 17 , b = 24 , c = 11 .
Therefore, a + b + c = 17 + 24 + 11 = 52 a + b + c = 17 + 24 + 11 = 52 a + b + c = 17 + 24 + 11 = 52 .
So the correct option is:
Option B: 52