Problem 30 - Algebra and Trigonometry

Question

Evaluate the expression without using a calculator. $$ \left(\sin \frac{\pi}{3} \cos \frac{\pi}{4}-\sin \frac{\pi}{4} \cos \frac{\pi}{3}\right)^{2} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{1}{2} - \frac{\sqrt{3}}{4} \).

Step 1: Recall Trigonometric Values

First, recall the values of the trigonometric functions for specific angles. - \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \)- \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Step 2: Simplify the Expression

Substitute the trigonometric values from Step 1 into the expression:\[\left( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \right)^2\]

Step 3: Calculate Each Term

Now calculate the multiplication for each term:- \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} \)- \( \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{4} \)

Step 4: Subtract the Results

Subtract the second term from the first term:\[\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\]

Step 5: Square the Result

Compute the square of the result from Step 4:\[\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2 = \frac{(\sqrt{6} - \sqrt{2})^2}{16}\]Use the identity \((a-b)^2 = a^2 - 2ab + b^2\):\[(\sqrt{6})^2 - 2(\sqrt{6})(\sqrt{2}) + (\sqrt{2})^2 = 6 - 2\sqrt{12} + 2\]

Step 6: Simplify the Result

Simplify \(6 - 2\sqrt{12} + 2\) and substitute back:\[6 + 2 = 8 \quad \text{and} \quad \sqrt{12} = 2\sqrt{3}\]Thus, \(-2\sqrt{12} = -4\sqrt{3}\), so the entire squared expression is:\[8 - 4\sqrt{3}\]

Step 7: Divide to Find Final Answer

Divide the simplified expression from Step 6 by 16:\[\frac{8 - 4\sqrt{3}}{16} = \frac{8}{16} - \frac{4\sqrt{3}}{16} = \frac{1}{2} - \frac{\sqrt{3}}{4}\]

Key Concepts

Sine and Cosine Values

Understanding trigonometric values is fundamental in trigonometry. Sine and cosine functions help us relate angles to the lengths of sides in a right triangle.

For this exercise, we use specific angle values:

• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \)
• \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

These values are essential because they form the basis of evaluating and simplifying the given trigonometric expression correctly. Knowing them by heart can greatly simplify solving trigonometric problems without a calculator.

To recall these values easily, remember that for angles \( \frac{\pi}{3} \) (or 60 degrees) and \( \frac{\pi}{4} \) (or 45 degrees), you can visualize an equilateral triangle and a square with diagonals, respectively. This visualization gives a geometric way to understand these trigonometric ratios.

Expression Simplification

The process of expression simplification involves transforming a complex expression into a more manageable form without changing its value. In trigonometry, applying correct identities and values is key to this.

As shown in the exercise, by substituting the known values, the expression:\[\left( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos \frac{\pi}{3} \right)^2\]becomes:\[\left( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \right)^2\]

Calculating each product and then subtracting gives:

• \( \frac{\sqrt{6}}{4} \) for \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \)
• \( \frac{\sqrt{2}}{4} \) for \( \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \)

Subtracting these, we get \( \frac{\sqrt{6} - \sqrt{2}}{4} \). This step-by-step approach effectively handles the complexity and leads us to the simplified expression.

Angle Subtraction Formulas

The angle subtraction formula is a key trigonometric identity that helps simplify expressions involving two angles. For sine, the angle subtraction formula is:\[\sin(a - b) = \sin a \cos b - \cos a \sin b\]

In our expression \( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos \frac{\pi}{3} \), we are essentially using this angle subtraction formula. This simplifies the problem by converting our original expression into a subtraction of products, aligning perfectly with the formula's structure.

The calculated expression corresponds to \( \sin(\frac{\pi}{3} - \frac{\pi}{4}) \), which would simplify directly to \( \sin \frac{\pi}{12} \), a less common angle where values might be derived or verified by fundamental identities. The subtraction formula hence offers a powerful tool for breaking down more complex trigonometric terms for deeper mathematical understanding.