Solving Trigonometric Expressions: The Relationship Between sin(A B) and cos(A-B)

Understanding the relationships between trigonometric functions is a fundamental skill in advanced mathematics. This article will explore the evaluation of a specific trigonometric expression using known values of cosine and tangent, and then delve into how the sin(A B) and cos(A - B) identities can be applied to simplify and solve the given problem. Let's consider the equation:

if cosA 1/2 and tanB 1/√3, what is the value of sin(A B) 2cos(A - B)?

Step-by-Step Solution

Step 1: Determine the Values of A and B

First, we need to find the values of angles A and B based on the given cosine and tangent values.

Considering the unit circle and trigonometric values, we can deduce:

Step 2: Apply Trigonometric Identities

Next, let's apply the trigonometric identities for sin and cos to find the values of sin(A B) and cos(A - B).

Sin(A B)

Using the identity sin(A B) sinAcosB cosAsinB:

Since A π/3 and B π/6:

sin(π/3 π/6) sin(π/2) 1

2cos(A - B)

Using the identity cos(A - B) cosAcosB sinAsinB:

cos(π/3 - π/6) cos(π/6) √3/2

Hence, 2cos(π/3 - π/6) 2 × (√3/2) √3

Step 3: Combine the Results

Lastly, we combine the results obtained from the identities to find the value of the given expression:

sin(A B) 2cos(A - B) 1 √3

Additional Tricks for Solving Trigonometric Expressions

Trigonometric Identity for sin(A B)

This identity can be very useful in solving more complex trigonometric expressions. For instance, knowing that:

sin(A B) sinAcosB cosAsinB

can help simplify expressions where angles are added together.

Trigonometric Identity for cos(A - B)

Similarly, the identity:

cos(A - B) cosAcosB sinAsinB

is essential for simplifying expressions involving the subtraction of angles.

Conclusion

By understanding the relationships between trigonometric functions and applying the relevant identities, we can solve complex trigonometric expressions with ease. In this article, we provided a detailed solution to the given expression, demonstrating the use of cosine and tangent values, and the application of trigonometric identities. This approach can be applied to various similar problems, making it a valuable skill for students and mathematicians alike.

For further exploration and practice, consider working through additional trigonometric expression problems that involve different trigonometric identities and values of sine, cosine, and tangent.