If the value of (3 cos 36° + 5 sin 18°) / (5 cos 36° - 3 sin 18°) is (a√5 - b) / c, where a, b, c ar

Q: Why do we memorize $\sin 18^\circ$ and $\cos 36^\circ$? Can't we just derive them using half-angle formulas?

While you absolutely can derive them, in a high-pressure environment like the JEE Advanced, time is your most valuable currency. Memorizing these specific values allows you to bypass several minutes of algebraic derivation, giving you a significant competitive edge. Think of it as having a shortcut in your mental toolkit.

Q: Why did we multiply by the conjugate in the final steps? Is there another way to simplify the expression?

Rationalizing the denominator using the conjugate is the standard, most robust method to eliminate irrational numbers from the denominator. It transforms the denominator into a rational integer, which is essential for matching the required form $\frac{a\sqrt{5}-b}{c}$. Without this step, you would be left with an irrational denominator, making it impossible to identify the values of $a, b,$ and $c$.

Q: What if I get a negative sign in the denominator after rationalizing? Does it matter?

It matters significantly! The problem asks for the form $\frac{a\sqrt{5}-b}{c}$, where $a, b,$ and $c$ are natural numbers. If your calculation results in a negative denominator, you must multiply both the numerator and the denominator by $-1$. This will shift the negative sign to the numerator, allowing you to correctly identify $a, b,$ and $c$ as positive integers.

Question

If the value of $\frac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ}$ is $\frac{a\sqrt{5}-b}{c}$, where $a, b, c$ are natural numbers and $\gcd(a, c) = 1$, then $a + b + c$ is equal to :

Options:

(A) 40

(B) 52

(C) 50

(D) 54

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem asks us to evaluate the following trigonometric expression:

$$E = \frac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ}$$

At first glance, this looks like a complex arrangement of radicals. However, the angles $18^\circ$ and $36^\circ$ are fundamental to the geometry of a regular pentagon and are intimately tied to the Golden Ratio.

### The Foundation: Standard Values

Before we dive into the algebra, we must utilize the standard values for these trigonometric functions. You should commit these to memory for competitive examinations:

$$\sin 18^\circ = \frac{\sqrt{5}-1}{4}$$

$$\cos 36^\circ = \frac{\sqrt{5}+1}{4}$$

Substituting these values directly into our expression, we obtain:

$$E = \frac{3\left(\frac{\sqrt{5}+1}{4}\right) + 5\left(\frac{\sqrt{5}-1}{4}\right)}{5\left(\frac{\sqrt{5}+1}{4}\right) - 3\left(\frac{\sqrt{5}-1}{4}\right)}$$

### The Elegant Simplification

We have a fraction within a fraction. To simplify this efficiently, notice that every term in the numerator and denominator shares a common denominator of $4$.

By multiplying both the numerator and the denominator by $4$, we clear the clutter instantly:

$$E = \frac{3(\sqrt{5}+1) + 5(\sqrt{5}-1)}{5(\sqrt{5}+1) - 3(\sqrt{5}-1)}$$

### The Algebraic Grind

Now, we expand the terms methodically. In the numerator, we distribute the constants:

$$3\sqrt{5} + 3 + 5\sqrt{5} - 5 = 8\sqrt{5} - 2$$

In the denominator, we must be careful with the signs:

$$5\sqrt{5} + 5 - 3\sqrt{5} + 3 = 2\sqrt{5} + 8$$

Our expression is now:

$$E = \frac{8\sqrt{5} - 2}{2\sqrt{5} + 8}$$

Factoring out a $2$ from both the numerator and the denominator, we simplify the expression to:

$$E = \frac{4\sqrt{5} - 1}{\sqrt{5} + 4}$$

### The Surgical Strike: Rationalization

To reach the final form, we must rationalize the denominator $\sqrt{5} + 4$. We multiply the numerator and denominator by the conjugate, $\sqrt{5} - 4$:

$$E = \frac{(4\sqrt{5} - 1)(\sqrt{5} - 4)}{(\sqrt{5} + 4)(\sqrt{5} - 4)}$$

The denominator becomes $(\sqrt{5})^2 - 4^2 = 5 - 16 = -11$. The numerator expands as follows:

$$(4\sqrt{5} - 1)(\sqrt{5} - 4) = 20 - 16\sqrt{5} - \sqrt{5} + 4 = 24 - 17\sqrt{5}$$

Thus, the expression is:

$$E = \frac{24 - 17\sqrt{5}}{-11} = \frac{17\sqrt{5} - 24}{11}$$

### The Final Triumph

We compare this to the required form $\frac{a\sqrt{5}-b}{c}$. By direct inspection, we identify the constants:

$$a = 17, \quad b = 24, \quad c = 11$$

The condition $\gcd(17, 11) = 1$ is satisfied. Finally, we calculate the sum:

**$$a + b + c = 17 + 24 + 11 = 52$$**

Animated Solution for Mathematics - Trigonometry: If the value of $\frac{3 c o s 3 6 ^{\circ} + 5 s i n 1 8 ^{\circ}}{5 c o s 3 6 ^{\circ} - 3 s i n 1 8 ^{\circ}}$ is $\frac{a 5 - b}{c}$ , where $a, b, c$ are natural numbers and $g cd (a, c) = 1$ , then $a + b + c$ is equal to :

Select Answer:

Visualized Solution

Standard Values of $sin 1 8^{\circ}$ and $cos 3 6^{\circ}$

Substitution into the Expression

Canceling the Common Denominator

Expanding the Numerator

Expanding the Denominator

Combining Like Terms

Factoring Out Common Terms

Rationalizing the Denominator

Multiplying the Numerators

Multiplying the Denominators

Adjusting the Negative Sign

Comparing with the Given Form

Final Calculation: $a + b + c$

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Foundation

The Elegant Simplification

The Algebraic Grind

The Surgical Strike

The Final Triumph

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