Simplifying Trigonometric Expressions to Find Exact Values: A Comprehensive Guide
Solving trigonometric expressions can often feel like a challenging task, but with the right approach, it becomes much more manageable. In this article, we will delve into the methods and techniques used to simplify complex trigonometric expressions and find their exact values. Specifically, we will explore a detailed example of how to evaluate a product of sines, and discuss the step-by-step process involved.
Understanding the Trigonometric Expression
Given the expression:
( sin{frac{37pi}{144}} cdot sin{frac{13pi}{72}} cdot sin{frac{35pi}{144}} cdot sin{frac{11pi}{72}} frac{1}{42} sin{frac{37pi}{144}} sin{frac{35pi}{144}} 2 sin{frac{13pi}{72}} sin{frac{11pi}{72}} frac{1}{4} cos{frac{pi}{72}} - cos{frac{pi}{2}} cos{frac{pi}{36}} - cos{frac{pi}{3}} )
Let's break this down step by step to find the exact value of the expression.
Step-by-Step Solution
We start by pairing the first and third factors together as well as the second and fourth factors together:
( P sin{frac{37pi}{144}} sin{frac{35pi}{144}} cdot sin{frac{13pi}{72}} sin{frac{11pi}{72}} )
Using the product-to-sum identity:
( sin{A} sin{B} frac{1}{2} left( cos{(A - B)} - cos{(A B)} right) )
We rewrite the product as:
( P frac{1}{2} left[ cos{left( frac{37pi}{144} - frac{35pi}{144} right)} - cos{left( frac{37pi}{144} frac{35pi}{144} right)} right] cdot frac{1}{2} left[ cos{left( frac{13pi}{72} - frac{11pi}{72} right)} - cos{left( frac{13pi}{72} frac{11pi}{72} right)} right] )
Which simplifies to:
( P frac{1}{4} cos{left(frac{pi}{72}right)} left[ cos{left(frac{pi}{36}right)} - frac{1}{2} right] )
Applying the product-to-sum identity again:
( P frac{1}{4} cos{left(frac{pi}{72}right)} left[ cos{left(frac{pi}{36} - frac{pi}{72}right)} cos{left(frac{pi}{36} frac{pi}{72}right)} - frac{1}{2} right] )
This further simplifies to:
( P frac{1}{8} cos{left(frac{pi}{24}right)} )
To find the exact value, we repeatedly apply the half-angle identity:
( cos{left( frac{A}{2} right)} sqrt{frac{1 cos{A}}{2}} )
We have:
( P frac{1}{8} sqrt{frac{1 cos{left(frac{2pi}{24}right)}}{2}} frac{1}{8} sqrt{frac{1 sqrt{frac{1 cos{left(frac{4pi}{24}right)}}{2}}}{2}} frac{1}{8} sqrt{frac{1 sqrt{frac{1 frac{sqrt{3}}{2}}{2}}}{2}} )
This simplifies to:
( P frac{1}{16} sqrt{2 sqrt{2 sqrt{3}}} approx 0.1239 )
Conclusion
In conclusion, we demonstrated a detailed approach to simplify a complex trigonometric expression and find its exact value. By using the product-to-sum identity and the half-angle identity, we were able to transform the initial expression into a simpler form and ultimately compute its exact value.