Exploring Trigonometric Identities: Sin(xy) When Cos(x) Cos(y)

In the realm of trigonometry, the relationships between various trigonometric functions provide a rich field for exploration and problem-solving. One such area involves the function sin(xy), particularly in scenarios where the cosine of two angles, cos(x) and cos(y), are equal. This article delves into the value of sin(xy) under these specific conditions, providing a comprehensive guide for students, educators, and enthusiasts alike.

Understanding the Basics

Let's start by quoting the given formula: sin(xy) sin(x)cos(y) sin(y)cos(x). This identity is a fundamental tool in trigonometry, connecting the sine and cosine functions in a multiplicative manner. When we encounter the condition cos(x) cos(y), it implies a symmetry in the system that can be exploited to simplify our calculations.

The Case: cos(x) cos(y)

When cos(x) cos(y), we can use the Pythagorean identity, which states that for any angle, sin^2(θ) cos^2(θ) 1. Applying this identity to our scenario, we find that sin(x) sin(y). This key observation allows us to simplify the expression for sin(xy).

Calculation for the Case cos(x) cos(y) and sin(x) sin(y)

Given that cos(x) cos(y) and sin(x) sin(y), we can substitute these into the formula for sin(xy). The formula becomes:

sin(xy) sin(x)cos(y) sin(y)cos(x)

Since sin(x) sin(y) and cos(x) cos(y), we can simplify this expression:

sin(xy) sin(x)cos(x) sin(x)cos(x)

sin(xy) 2sin(x)cos(x)

Using the double-angle formula for sine, sin(2x) 2sin(x)cos(x), we find that:

sin(xy) sin(2x)

Calculation for the Case cos(x) cos(y) and sin(x) -sin(y)

However, if sin(x) -sin(y) while cos(x) cos(y), the scenario changes slightly. We substitute these conditions into the formula for sin(xy):

sin(xy) sin(x)cos(y) sin(y)cos(x)

Since sin(x) -sin(y) and cos(x) cos(y), we get:

sin(xy) (-sin(y))cos(y) (sin(y))cos(y)

sin(xy) -sin(y)cos(y) sin(y)cos(y)

sin(xy) 0

Conclusion

In conclusion, the value of sin(xy) when cos(x) cos(y) can be determined by considering the specific relationship between sin(x) and sin(y). When both sine values are equal (sin(x) sin(y)), the value of sin(xy) is sin(2x). However, if the sine values are negatives of each other (sin(x) -sin(y)), the value of sin(xy) is zero.

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Further Reading

For a deeper understanding of trigonometric identities and their applications, you may wish to explore:
- Trigonometric Identities and Examples
- Applications of Trigonometry in Real Life
- Solving Trigonometric Equations