Simplifying Trigonometric Expressions: 1 sin2x

Understanding and Simplifying Trigonometric Expressions

In trigonometry, simplifying expressions is an essential skill. This is particularly true when dealing with identities. One such expression that often arises in trigonometric problems is 1 sin2x. Let's explore its simplification step by step, providing insights and verifying our answer.

First, recall a fundamental trigonometric identity: cos2x sin2x 1. From this, we can derive several useful forms, including sin2x 1 - cos2x. Utilizing this, we can express the original expression in different ways.

The Process of Simplification

Starting with the given expression: 1 sin2x

Step 1: Substituting the Identity

We know from the trigonometric identity that sin2x 1 - cos2x. Therefore, substituting this into our expression:

1   sin2x  1   (1 - cos2x)

Note 1: The parentheses indicate that the expression inside should be fully computed before adding 1 to it.

Step 2: Distributive Property

Next, we apply the distributive property of addition and subtraction inside the parentheses:

1   (1 - cos2x)  1   1 - cos2x

By simplifying the expression, we get:

2 - cos2x

Step 3: Validating the Simplification

To verify our solution, let's consider some example values for x. We'll use (x frac{pi}{4}) and substitute it into both the original and simplified expressions.

Original Expression: 1 sin2x

If (x frac{pi}{4}), then: Calculate (2x frac{pi}{2}) Find (sin(frac{pi}{2}) 1) Hence, (1 sin(frac{pi}{2}) 1 1 2)

Simplified Expression: 2 - cos2x

If (x frac{pi}{4}), then: Calculate (2x frac{pi}{2}) Find (cos(frac{pi}{2}) 0) Hence, (2 - cos(frac{pi}{2}) 2 - 0 2)

Both expressions evaluate to 2 when (x frac{pi}{4}), confirming our simplification is accurate.

Conclusion

In conclusion, the trigonometric expression 1 sin2x simplifies to 2 - cos2x using the trigonometric identity sin2x 1 - cos2x. This simplification can be useful in various trigonometric contexts, including proving identities and solving equations.

Key Takeaways

sin2x 1 - cos2x The distributive property simplifies expressions. Verification with example values ensures accuracy.

By mastering these skills, you will be better equipped to tackle more complex trigonometric problems. If you have any further questions or need clarification, feel free to ask.