Understanding Trigonometric Identities: Evaluating Sin21 cos69 sin69 cos21
In this article, we will delve into the evaluation of the trigonometric expression sin21° cos69° sin69° cos21°. We will explore how to use the sine addition formula to simplify and solve this expression. This will be useful for anyone studying trigonometry, as it introduces key concepts of trigonometric identities and their applications.
The Sine Addition Formula
One of the fundamental trigonometric identities is the sine addition formula:
sinarlm; bsina cosb?cosa sinb.
Simplifying the Expression
Given the expression sin21° cos69° sin69° cos21°, we can see that it is already in a form that can be solved using the sine addition formula. Let us apply this identity step by step:
sin21° cos69° sin69° cos21° can be rewritten using the sine addition formula as follows:
sin(21° 69°) 1
This simplification is possible because 21° 69° 90°, and we know that sin90° 1.
Thus, the final value of the expression sin21° cos69° sin69° cos21° is 1.
Additional Notes on Trigonometric Identities
Understanding and applying trigonometric identities is crucial for various mathematical and scientific applications. Here are a few additional notes:
Sine of a Sum: The general form of the sine addition formula is sin(a b) sina cosb ? cosa sinb. Evaluating Specific Angles: For specific angles such as 90°, the sine value is always 1, which simplifies many trigonometric calculations significantly. Applications: Trigonometric identities are used in fields such as physics, engineering, and calculus, particularly in solving complex problems involving periodic functions and wave forms.Conclusion
By leveraging the sine addition formula, we have successfully evaluated the expression sin21° cos69° sin69° cos21° to find its value as 1. This exercise demonstrates the power and utility of trigonometric identities in simplifying and solving trigonometric expressions. Understanding these identities and their applications can greatly enhance one's problem-solving skills in mathematics and related fields.