Simplifying Complex Trigonometric Expressions: sin 60° cos 30° / (sin 45° × tan 30° × sec 45°)

In this article, we will walk through the process of evaluating the expression:

[frac{sin 60^circ cos 30^circ}{sin 45^circ times tan 30^circ times sec 45^circ}]

Step 1: Understanding Trigonometric Values

To solve the given expression, we need to recall the values of several trigonometric functions for specific angles. Here are the values we will use:

[sin 60^circ frac{sqrt{3}}{2}] [cos 30^circ frac{sqrt{3}}{2}] [sin 45^circ frac{sqrt{2}}{2}] [tan 30^circ frac{1}{sqrt{3}}] [sec 45^circ sqrt{2}]

Step 2: Substituting the Values

Now, we will substitute these values into the expression and simplify step by step.

Numerator

The numerator of the expression is:

[sin 60^circ cos 30^circ frac{sqrt{3}}{2} cdot frac{sqrt{3}}{2} frac{3}{4} cdot sqrt{3} sqrt{3}]

Denominator

The denominator of the expression is:

[sin 45^circ times tan 30^circ times sec 45^circ frac{sqrt{2}}{2} cdot frac{1}{sqrt{3}} cdot sqrt{2} frac{sqrt{2} cdot sqrt{2}}{2sqrt{3}} frac{2}{2sqrt{3}} frac{1}{sqrt{3}}]

Step 3: Simplifying the Final Expression

Now, we can combine the results by dividing the numerator by the denominator:

tttttttttttttt tttttttttttttt tttttttttttttttmsqrt ttttttttttttttt3 tttttttttttttt tttttttttttttt/ tttttttttttttt"(frac{sqrt{3}}{frac{1}{sqrt{3}}}) tttttttttttttt tttttttttttttt ttttttttttttttt3 tttttttttttttt tttttttttttttt* tttttttttttttt ttttttttttttttt3 tttttttttttttt tttttttttttttt tttttttttttttt3 tttttttttttttt tttttttttttttt

Thus, the final answer is:

boxed{3}

Purpose and Importance of Trigonometric Identities

Understanding and applying trigonometric identities is crucial in solving complex expressions. These identities can help simplify terms, making calculations more manageable and accurate. Familiarity with these identities is particularly useful in advanced mathematics, physics, and engineering.

Key Takeaways

Recall and apply standard trigonometric values for common angles. Use algebraic manipulation to simplify expressions involving trigonometric functions. Verify the correctness of each step in the calculation for clarity and accuracy.

Conclusion

By following the steps outlined in this article, you have successfully simplified the expression sin 60° cos 30° / (sin 45° × tan 30° × sec 45°) to 3. This example demonstrates the importance of trigonometric identities in evaluating complex trigonometric expressions and highlights the value of understanding these fundamental mathematical tools.