Problem 30 - Precalculus Mathematics for Calculus

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 value is \(\frac{1}{2} - \frac{\sqrt{3}}{4}\).

Step 1: Apply the angle subtraction formula

Identify the expression inside the square, which is \( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos \frac{\pi}{3} \). Recognize that this matches the sine angle subtraction formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Here, \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \). Therefore, the expression can be rewritten as \( \sin(\frac{\pi}{3} - \frac{\pi}{4}) \).

Step 2: Simplify the angle

Simplify the angle expression by subtracting \( \frac{\pi}{4} \) from \( \frac{\pi}{3} \). This gives \( \sin(\frac{\pi}{3} - \frac{\pi}{4}) = \sin(\frac{4\pi}{12} - \frac{3\pi}{12}) = \sin\frac{\pi}{12} \).

Step 3: Recognize the need for exact trigonometric identity

Since \( \frac{\pi}{12} \) is not one of the standard angles for which we know a simple trigonometric value (like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \)), use known trigonometric sums or half-angle formulas, recognizing its approximate nature in exact analytical solutions: \( \sin\frac{\pi}{12} \) approximates to known values based on trigonometric identity simplifications (e.g. \( \sin\frac{\pi}{12} = \sin(15^o) = \frac{\sqrt{6} - \sqrt{2}}{4} \) if derived through trigonometric identities).

Step 4: Compute the squared result

Square the simplified trigonometric expression: \( (\sin\frac{\pi}{12})^2 \). Using \( \sin\frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4} \), square this value to obtain \( \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2 = \frac{(\sqrt{6} - \sqrt{2})^2}{16} \).

Step 5: Simplify the result of squaring

Calculate \((\sqrt{6} - \sqrt{2})^2 = 6 - 2\sqrt{12} + 2 = 8 - 2\sqrt{12}\). Simplify further to \(8 - 4\sqrt{3}\), and divide by 16: \( \frac{8 - 4\sqrt{3}}{16} = \frac{1}{2} - \frac{\sqrt{3}}{4} \).

Key Concepts

Sine Angle Subtraction Formula

The sine angle subtraction formula is a useful tool in trigonometry that allows us to express the sine of a difference of two angles in terms of the sines and cosines of the individual angles. This formula is given by:\[\sin(a - b) = \sin a \cos b - \cos a \sin b\]
In this specific exercise, we have the expression \( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos \frac{\pi}{3} \).
By comparing this to the sine angle subtraction formula, we identify \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \).

• This recognition enables us to rewrite the expression as \( \sin(\frac{\pi}{3} - \frac{\pi}{4}) \).
• This step is essential as it simplifies the original problem by transforming it into a single sine expression.

Once we rewrite the problem, we need to proceed with calculating the exact sine value of the new angle. This technique simplifies solving complex trigonometric problems by breaking them down into manageable parts.

Exact Trigonometric Values

Determining exact trigonometric values can often involve exploring well-known identities and values derived from specific angles known through the unit circle.
Some well-known angles include \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), etc. These angles have known sine, cosine, and tangent values that allow us to work without a calculator.

• For \( \sin\left(\frac{\pi}{12}\right) \), we cannot rely on standard angles, hence we must deep dive into trigonometric identities.

For example, knowing that \( \sin\left(\frac{\pi}{12}\right) = \sin(15^o) \) and calculating its value using:
\[ \sin\left(15^o\right) = \frac{\sqrt{6} - \sqrt{2}}{4} \]
This derivation often involves using additional transformations and angle identities like sum, difference, or even half-angle formulas.

Though complex, these techniques ensure that we obtain the exact value needed, maintaining precision in trigonometric expressions. Understanding these derivations aids in solving problems that go beyond the typical angles seen in basic trigonometry.

Trigonometric Simplification

Simplification in trigonometry is about reducing expressions to their simplest forms. It involves using identities, algebraic manipulation, and sometimes, approximations.
Once we've determined the exact trigonometric value, the simplification process typically involves algebra. In this problem, simplifying \( \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2 \) entails squaring the expression:

• Calculate \((\sqrt{6} - \sqrt{2})^2\).
• The steps show it equals \(6 - 2\sqrt{12} + 2 = 8 - 4\sqrt{3}\).

Further simplification requires you to

• divide the whole expression by 16, resulting in \[ \frac{1}{2} - \frac{\sqrt{3}}{4} \].

This final value is the minimal expressional form of the square, demonstrating thorough mathematical precision and conciseness.
Understanding and practicing simplifications help significantly in handling more advanced trigonometric problems effectively.