Problem 28 - 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

\(\left(\sin\frac{\pi}{3} \cos\frac{\pi}{4} - \sin\frac{\pi}{4} \cos\frac{\pi}{3}\right)^2 = \frac{1}{2} - \frac{\sqrt{3}}{4}\)."

Step 1: Recall Trigonometric Values

First, recall the required trigonometric values. \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\), \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), and \(\cos \frac{\pi}{3} = \frac{1}{2}\).

Step 2: Substitute and Simplify Inside the Expression

Substitute these values into the expression \(\left( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos \frac{\pi}{3} \right)^{2}\).Calculate: \(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}\).

Step 3: Evaluate Each Part of the Calculation

First, calculate \(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}\).Next, calculate \(\frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{4}\).

Step 4: Subtract the Results

Subtract the two results from Step 3:\(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Step 5: Square the Result

Square the result from Step 4:\(\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2 = \frac{(\sqrt{6} - \sqrt{2})^2}{16}\).

Step 6: Expand and Simplify the Squared Expression

Expand \((\sqrt{6} - \sqrt{2})^2 = 6 - 2 \times \sqrt{6} \times \sqrt{2} + 2\).Simplify this to obtain \(6 - 2\sqrt{12} + 2\).Further simplify \(2\sqrt{12}\) to \(4\sqrt{3}\).

Step 7: Final Simplification

The expanded terms become \(6 + 2 - 4\sqrt{3} = 8 - 4\sqrt{3}\).Divide by 16 to simplify the term: \(\frac{8 - 4\sqrt{3}}{16} = \frac{1}{2} - \frac{\sqrt{3}}{4}\).Conclude that \(\left(\sin \frac{\pi}{3} \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos \frac{\pi}{3}\right)^{2} = \frac{1}{2} - \frac{\sqrt{3}}{4}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are often utilized in various applications such as geometry, physics, and engineering.
They allow us to explore the relationships between the angles and sides of triangles. The primary trigonometric functions are sine ( \( \sin \) ), cosine ( \( \cos \) ), and tangent ( \( \tan \) ). These are crucial in analyzing right-angled triangles.

• The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse.
• Cosine is the ratio of the adjacent side to the hypotenuse.
• Tangent is the ratio of the opposite side to the adjacent side.

These definitions extend beyond right-angled triangles, describing periodic waveforms and oscillations. Other functions derived from these primary ones include secant, cosecant, and cotangent. Understanding these relationships is the cornerstone of trigonometry.

Sine and Cosine

Sine and cosine are the two most widely used trigonometric functions that describe a circular motion or oscillation. These functions help us understand and predict behaviors in circular geometry and wave patterns.
The function sine ( \( \sin \) ) deals primarily with the vertical component or height of an angle. Cosine ( \( \cos \) ) relates to its horizontal component.Imagine a point moving around a circle that's based around a coordinate system. The x-coordinate of this point is represented by the cosine of the angle, whereas the y-coordinate is represented by the sine.
For example:

• At \( \pi /4 \) radians, both sine and cosine have equal values of \( \sqrt{2}/2 \) pointing to a 45-degree angle having equal x and y coordinates.
• At \( \pi /2 \) radians, sine equals 1, while cosine equals 0, aligning with the vertical peak of a circle.

These concepts are not restricted to right angles or even angles within 0 to 90 degrees but can extend infinitely within continuous wave functions.

Angle Values in Radians

Understanding angle measurements in radians is essential in mathematics, particularly in trigonometry where angle values are often presented. Radians provide a more natural way of measuring angles compared to degrees because they relate directly to the arc length of a circle.
A radian is defined as the angle created by wrapping the radius of a circle along its circumference. Thus, one complete rotation around a circle (360 degrees) corresponds to \( 2\pi \) radians.
For frequently used angles:

• \( \pi/3 \) radians is equivalent to 60 degrees.
• \( \pi/4 \) radians is equivalent to 45 degrees.
• \( \pi/2 \) radians corresponds to 90 degrees.

Utilizing radians simplifies computations, especially when working with trigonometric functions or calculus, as many formulas and mathematical constants feature naturally in radians. Understanding the conversion between degrees and radians is crucial to mastering trigonometry.