Problem 29 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4} $$

Step-by-Step Solution

Answer

The exact value of the expression is \( \frac{1}{2} \).

Step 1: Identify the identity

The given expression resembles the right-side of the sine difference identity. This identity is denoted as \( \sin(A-B) = \sin A \cos B - \cos A \sin B \).

Step 2: Apply the identity

Applying this identity to the provided equation, we get: \( \sin(\frac{5\pi}{12} - \frac{\pi}{4}) \).

Step 3: Simplify the expression

Simplifying the expression further, \(\frac{5\pi}{12} - \frac{\pi}{4} = \frac{5\pi}{12} - \frac{3\pi}{12} = \frac{2\pi}{12}\). This can be simplified further to \(\frac{\pi}{6}\).

Step 4: Find exact value

Now, the simplified expression becomes \( \sin(\frac{\pi}{6}) \). To find its exact value, we refer to the unit circle or a table of trigonometric values to find that \( \sin(\frac{\pi}{6}) = \frac{1}{2} \).

Key Concepts

Trigonometric Identities

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. At the heart of this field are trigonometric identities, which are equations used to relate various trigonometric functions to one another. These identities are vital for simplifying complex expressions and solving equations. One example, as seen in the exercise, is the sine difference formula.

Understanding trigonometric identities is crucial for students, not just for solving specific problems, but also for gaining insight into the structure of mathematics and the symmetries of geometry. They are the foundation for advanced topics in calculus, physics, and engineering. By internalizing these identities, students can more easily grasp the intrinsic patterns of trigonometry.

Sine Difference Formula

The sine difference formula is a specific type of trigonometric identity used to express the sine of the difference between two angles. The formula is written as \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \.\] This formula allows for the simplification of trigonometric expressions and is especially useful when calculating the sine of an angle that's not readily available on the unit circle.

Such formulas not only ease the complexity of computations but also help to understand how different trigonometric values interact with each other. When approached methodically, as demonstrated in the exercise, the sine difference formula can transform a seemingly arduous problem into a straightforward task, revealing an exact trigonometric value.

Exact Trigonometric Values

Exact trigonometric values refer to the precise, fractional, or radical values of trigonometric functions for specific angles. These values are essential for precise calculations in mathematics, science, and engineering. For instance, the exercise requires the exact value of \( \sin(\frac{\pi}{6}) \), which is \( \frac{1}{2} \).

The ability to determine these exact values is important as it can enhance the accuracy of computations in geometric problems, waveform analysis, and more. Students can find these values using the unit circle, a table of common angles, or by memorizing the most frequently used exact values. It's a perfect illustration of how mathematical concepts are interconnected -- the unit circle and trigonometric identities converge to produce these exact values.

Unit Circle

The unit circle is a fundamental tool in trigonometry, defined as a circle with a radius of one, centered at the origin of a two-dimensional coordinate system. It is used to visually and conceptually represent trigonometric functions. Angles on the unit circle correspond to points on its circumference, and the coordinates of these points align with the values of sine and cosine for that angle.

For instance, the angle \( \frac{\pi}{6} \) from the exercise corresponds to a point on the circle with coordinates \( (\frac{\sqrt{3}}{2}, \frac{1}{2}) \), which directly gives us the cosine and sine of the angle, respectively. This reiterates the circle's utility in deriving exact trigonometric values. By mastering the unit circle, students can easily bridge the gap between theoretical trigonometry and practical application.