Using Trigonometric Identities to Simplify and Prove Logarithmic Expressions
Understanding and applying trigonometric identities is a fundamental skill in mathematics, especially when manipulating trigonometric functions in expressions involving logarithms. In this article, we will explore how to utilize these identities to prove the equivalence between two logarithmic expressions involving trigonometric functions. Specifically, we will show that (frac{1}{3}ln tanfrac{3x}{2} frac{1}{3}ln csc 3x - cot 3x). This involves a step-by-step breakdown of the manipulation of logarithmic and trigonometric functions.
Step-by-Step Proving the Identity
The first step in proving the given identity is to start with the left-hand side of the equation, and then simplify it to match the right-hand side. We begin with the expression:
Left Hand Side
Expression
[ frac{1}{3} ln tan frac{3x}{2} ]Step 1: Express (tan frac{3x}{2}) in terms of sine and cosine
[ frac{1}{3} ln frac{sin frac{3x}{2}}{cos frac{3x}{2}} ]Step 2: Multiply numerator and denominator by 2 to use double angle identities
[ frac{1}{3} ln frac{2 sin^2 frac{3x}{2}}{2 sin frac{3x}{2} cos frac{3x}{2}} ]Step 3: Use double angle identities for sine and cosine
[ frac{1}{3} ln frac{1 - cos 3x}{sin 3x} ]Step 4: Rewrite using properties of logarithms
[ frac{1}{3} ln left(frac{1}{sin 3x} - frac{cos 3x}{sin 3x}right) ]Step 5: Simplify using (csc 3x frac{1}{sin 3x}) and (cot 3x frac{cos 3x}{sin 3x})
[ frac{1}{3} ln csc 3x - tanh 3x ]This matches exactly with the right-hand side of the equation, proving the identity:
Alternative Approach
Another way to approach the problem is to use the identity for tangent, by expressing (tanfrac{3x}{2}) in terms of sine and cosine, and simplifying using trigonometric identities. Specifically:
Using Trigonometric Identities Directly
Step 1: Express (tan frac{3x}{2}) in terms of sine and cosine
[ tan frac{3x}{2} frac{sin frac{3x}{2}}{cos frac{3x}{2}} ]Step 2: Multiply the numerator and the denominator by (cos frac{3x}{2})
[ tan frac{3x}{2} frac{sin frac{3x}{2} cos frac{3x}{2}}{cos^2 frac{3x}{2}} ]Step 3: Use the identity (cos^2 frac{3x}{2} 1 - sin^2 frac{3x}{2})
[ tan frac{3x}{2} frac{sin frac{3x}{2}}{1 - sin^2 frac{3x}{2}} cdot cos frac{3x}{2} ]Step 4: Rewrite the expression using (sin 3x 3 sin x - 4 sin^3 x)
[ tan frac{3x}{2} frac{2 sin frac{3x}{2}}{cos 3x} ]Step 5: Use the identity (sin frac{3x}{2} sin 3x cdot frac{1}{cos 3x})
[ tan frac{3x}{2} cot 3x cdot csc 3x ]Thus, we can write:
Final Simplification
[ frac{1}{3} ln tan frac{3x}{2} frac{1}{3} ln (cot 3x cdot csc 3x) ] [ frac{1}{3} (ln cot 3x ln csc 3x) ] [ frac{1}{3} ln cot 3x - frac{1}{3} ln csc 3x ] [ frac{1}{3} ln csc 3x - cot 3x ]This confirms the original identity, and provides an alternative route to the proof.
Key Trigonometric Identities Used
The process above illustrates the use of several key trigonometric identities:
Double Angle Identities: (sin 2theta 2 sin theta cos theta) Pythagorean Identity: (cos^2 theta sin^2 theta 1) Sum and Difference Identities: (sin(A B) sin A cos B cos A sin B) Product-to-Sum Identities: (sin A cos B frac{1}{2} [sin (A B) sin (A - B)])Understanding these identities and how to apply them is crucial in simplifying and solving complex trigonometric expressions.
Conclusion
In conclusion, we have successfully demonstrated the equivalence of the given logarithmic expressions using trigonometric identities. The process involves a series of simplifications and transformations, ultimately confirming that (frac{1}{3} ln tan frac{3x}{2} frac{1}{3} ln csc 3x - cot 3x). Understanding and applying these identities are fundamental in solving a wide range of trigonometric and logarithmic problems.
Additional Resources
For further learning, you may want to explore additional resources on trigonometric identities and logarithms. Some useful online resources and books include:
Trigonometry by Lial, Hornsby, Schneider, and Daniels Paul's Online Math Notes: Trigonometric Functions Wolfram Alpha for step-by-step verification of trigonometric identities