Question

The value of $\sin 18^{\circ}$ is

• A. 5 - 15
• B. 5 - 14
• C. 5 + 14
• D. 5 + 15

Hint:

$\sin 18^{\circ}$ is a very famous value in coordinate geometry and trigonometry. Memorizing $\sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$ along with its partner $\cos 36^{\circ} = \frac{\sqrt{5}+1}{4}$ helps bypass long quadratic derivations during tests!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem requires us to derive or recall the exact trigonometric value of $\sin 18^{\circ}$.

Step 2: Detailed Explanation:
Let's assign a variable to our target angle: $A = 18^{\circ}$. Multiplying both sides by 5: $$ 5A = 90^{\circ} \implies 2A + 3A = 90^{\circ} $$ Rearranging to separate the terms: $$ 2A = 90^{\circ} - 3A $$ Taking the sine function on both sides of the equation: $$ \sin(2A) = \sin(90^{\circ} - 3A) $$ Using the complementary trigonometric reduction identity $\sin(90^{\circ} - \theta) = \cos\theta$: $$ \sin 2A = \cos 3A $$ Applying the double-angle and triple-angle identity formulas ($\sin 2A = 2\sin A\cos A$ and $\cos 3A = 4\cos^3 A - 3\cos A$): $$ 2\sin A\cos A = 4\cos^3 A - 3\cos A $$ Since $A = 18^{\circ}$, $\cos 18^{\circ} \neq 0$. We can safely divide out $\cos A$ from both sides: $$ 2\sin A = 4\cos^2 A - 3 $$ Replace $\cos^2 A$ with $1 - \sin^2 A$ using the fundamental identity: $$ 2\sin A = 4(1 - \sin^2 A) - 3 $$ $$ 2\sin A = 4 - 4\sin^2 A - 3 \implies 4\sin^2 A + 2\sin A - 1 = 0 $$ This is a standard quadratic equation in terms of $\sin A$. Applying the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $$ \sin A = \frac{-2 \pm \sqrt{2^2 - 4(4)(-1)}}{2(4)} = \frac{-2 \pm \sqrt{4 + 16}}{8} = \frac{-2 \pm \sqrt{20}}{8} $$ Simplifying the radical term $\sqrt{20} = 2\sqrt{5}$: $$ \sin A = \frac{-2 \pm 2\sqrt{5}}{8} = \frac{-1 \pm \sqrt{5}}{4} $$ Since $18^{\circ}$ lies entirely inside the first quadrant, its sine value must be strictly positive. Therefore, we discard the negative root: $$ \sin 18^{\circ} = \frac{\sqrt{5} - 1}{4} $$

Step 3: Final Answer:
The exact value of $\sin 18^{\circ}$ is 5 - 14, corresponding to option (B).