Trigonometry & Inverse Trigonometry
Exact Values — cos 36°, sin 18°
nta_pyq_2024_apr
Grade 11
Question:
If the value of $\dfrac{3\cos36^\circ+5\sin18^\circ}{5\cos36^\circ-3\sin18^\circ}$ is $\dfrac{a\sqrt{5}-b}{c}$, where $a,b,c$ are natural numbers and $\gcd(a,c)=1$, then $a+b+c$ is equal to:
Step-by-Step Solution
Key Concept: Use $\cos36^\circ=\dfrac{\sqrt{5}+1}{4}$ and $\sin18^\circ=\dfrac{\sqrt{5}-1}{4}$. Numerator $=\dfrac{3(\sqrt{5}+1)+5(\sqrt{5}-1)}{4}=\dfrac{8\sqrt{5}-2}{4}$. Denominator $=\dfrac{5(\sqrt{5}+1)-3(\sqrt{5}-1)}{4}=\dfrac{2\sqrt{5}+8}{4}$.
$a=17,b=24,c=11$. $a+b+c=52$.
Correct Answer: 2