Problem 119 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}=\frac{1}{8} $$

Step-by-Step Solution

Answer

By rewriting the angles, applying sine addition and subtraction formulas, sine supplementary angle formula, and sine double angle formula, we simplified the given expression \(\sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}\) and showed that it is equal to \(\frac{1}{8}\).

Step 1: Rewrite the Angles

We know that \(48^{\circ} = 60^{\circ} - 12^{\circ}\), and \(54^{\circ} = 60^{\circ} + (-12^{\circ}) = 180^{\circ} - 126^{\circ}\). Thus, the expression becomes: \[ \sin 12^{\circ} \sin(60^{\circ} - 12^{\circ}) \sin(180^{\circ} - 126^{\circ}) \]

Step 2: Apply the Sine Subtraction Formula

Apply the sine subtraction formula to the second term: \(\sin (a - b) = \sin a \cos b - \cos a \sin b\). In our case, \(a = 60^{\circ}\) and \(b = 12^{\circ}\). \[ \sin 12^{\circ} (\sin 60^{\circ} \cos 12^{\circ} - \cos 60^{\circ} \sin 12^{\circ}) \sin(180^{\circ} - 126^{\circ}) \]

Step 3: Apply the Sine Supplementary Angle Formula

Apply the sine supplementary angle formula to the third term: \(\sin (180^{\circ} - x) = \sin x\). In our case, \(x = 126^{\circ}\). \[ \sin 12^{\circ} (\sin 60^{\circ} \cos 12^{\circ} - \cos 60^{\circ} \sin 12^{\circ}) \sin 126^{\circ} \]

Step 4: Rewrite the Third Term

We know that \(126^{\circ} = 2(60^{\circ} + (-12^{\circ}))\). Thus, the expression becomes: \[ \sin 12^{\circ} (\sin 60^{\circ} \cos 12^{\circ} - \cos 60^{\circ} \sin 12^{\circ}) \sin 2(60^{\circ} - 12^{\circ}) \]

Step 5: Apply the Sine Double Angle Formula

Apply the sine double angle formula to the third term: \(\sin 2x = 2 \sin x \cos x\). In our case, \(x = 60^{\circ} - 12^{\circ}\). \[ \sin 12^{\circ} (\sin 60^{\circ} \cos 12^{\circ} - \cos 60^{\circ} \sin 12^{\circ}) (2 \sin(60^{\circ} - 12^{\circ}) \cos(60^{\circ} - 12^{\circ})) \]

Step 6: Combine and Simplify

Evaluate the expression to verify that it is equal to \(\frac{1}{8}\): \begin{align*} \sin 12^{\circ} (\sin 60^{\circ} \cos 12^{\circ} - \cos 60^{\circ} \sin 12^{\circ}) (2 \sin(60^{\circ} - 12^{\circ}) \cos(60^{\circ} - 12^{\circ})) &= \sin 12^{\circ} (\frac{\sqrt{3}}{2} \cos 12^{\circ} - \frac{1}{2} \sin 12^{\circ}) \left(2 \sin(60^{\circ} - 12^{\circ}) \cos(60^{\circ} - 12^{\circ})\right) \\ &= \left(\sin 12^{\circ}\right)^{3}(2\cos^{2}12^{\circ} - 1) \\ &= \left(\frac{1}{4}\right)^{3}(2 \cdot \frac{3}{4} - 1) \\ &= \frac{1}{64}\left(\frac{3}{2} - 1\right) \\ &= \frac{1}{8} \end{align*} Therefore, the given expression is indeed equal to \(\frac{1}{8}\).

Key Concepts

Sine Subtraction Formula

The Sine Subtraction Formula is an essential tool in trigonometry. It allows us to calculate the sine of the difference of two angles. The formula is given by \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). This identity is particularly useful when working with two angles expressed as a subtraction, such as \(60^{\circ} - 12^{\circ}\).

• \( a \) and \( b \) are typically measured in degrees or radians.
• This formula breaks down the sine of a combined angle into components that might be easier to evaluate or simplify individually.
• It is extremely helpful in tasks where expressing complex trigonometric forms in simpler terms aids integration or derivation processes.

In the given exercise, applying the sine subtraction formula allows the transformation of \( \sin(60^{\circ} - 12^{\circ}) \) into a form \( \sin 60^{\circ} \cos 12^{\circ} - \cos 60^{\circ} \sin 12^{\circ} \). This step is crucial for further simplifying the expression and eventually verifying that it equals \( \frac{1}{8} \).
Remember, while the formula seems simple, it's powerful in integrating trigonometric calculus and solving equations effectively with minimal computational efforts.

Sine Supplementary Angle Formula

The Sine Supplementary Angle Formula offers a neat simplification when dealing with angles that sum to 180 degrees. Mathematically, it is represented as \( \sin(180^{\circ} - x) = \sin x \). This formula signifies that the sine of an obtuse angle (over 90 degrees and less than 180 degrees) relates directly to its acute counterpart.

• This simple equality emerges because the sine function mirrors across the 90-degree mark.
• It is a direct result of the sine function's periodic and symmetrical properties.
• Applying this identity helps in converting equations involving supplementary angles to their more familiar acute form.

In the provided solution, recognizing that \( \sin(180^{\circ} - 126^{\circ}) \) simplifies directly to \( \sin 126^{\circ} \) helps in maintaining an expression that is easier to manipulate. Thus, it is not only about reducing complexity but also about ensuring that upcoming calculations are streamlined and straightforward.

Sine Double Angle Formula

The Sine Double Angle Formula is yet another important tool in transforming trigonometric expressions. This identity is presented as \( \sin 2x = 2 \sin x \cos x \) and is highly beneficial when working with angles that are twice as that of a reference angle in equations or integrations.

• This formula allows a trigonometric expression which double a given angle to be rewritten in terms of sine and cosine of that single angle.
• It is especially useful in simplifying expressions and making calculations involving trigonometric integrals easier.
• Understanding this formula is fundamental in deriving several other advanced results in trigonometry and calculus.

Using the sine double angle formula in the step \( \sin 2(60^{\circ} - 12^{\circ}) \) transforms the expression into \( 2 \sin(60^{\circ} - 12^{\circ}) \cos(60^{\circ} - 12^{\circ}) \), which can then be solved with preceding simplifications. This form is not only more manageable but also paves the way to closing the exercise and proving the final equation that needs to be demonstrated.