Understanding the Trigonometric Identity for the Difference of Angles: sin(A - B) sin A cos B - sin B cos A

The trigonometric identity for the sine of the difference of two angles, sin(A - B), is a fundamental concept in trigonometry. This identity plays a crucial role in various applications, including calculus, physics, and engineering. In this article, we will delve into the derivation and application of this identity, providing clear explanations and practical examples.

Derivation of sin(A - B)

The identity for sin(A - B) is given by:

sin(A - B) sin A cos B - cos A sin B

To understand why this identity is true, we can start by considering the unit circle and the geometric interpretation of trigonometric functions. Alternatively, we can derive it using the angle sum and difference identities.

Geometric Interpretation

Consider two angles, A and B, on the unit circle. The coordinates of points representing these angles are:

text{Point for angle A:} (cos A, sin A) text{Point for angle B:} (cos B, sin B)

The point representing the angle (A - B) can be derived from the point (B) by rotating it by angle A. Using the vector addition, the coordinates of the point for angle (A - B) can be described by:

(cos(A - B), sin(A - B)) (cos B cos A sin B sin A, -cos B sin A sin B cos A)

This is derived by considering the transformation of coordinates under a rotation by A degrees.

Using Angle Sum and Difference Identities

The angle sum identities for sine and cosine are:

sin(A B) sin A cos B cos A sin B

cos(A B) cos A cos B - sin A sin B

By setting B to -B in the identity for sin(A B), we can obtain the identity for sin(A - B):

sin(A - B) sin A cos(-B) cos A sin(-B)

Using the properties of trigonometric functions, we know that:

cos(-B) cos B

sin(-B) -sin B

Substituting these into the equation, we get:

sin(A - B) sin A cos B cos A (-sin B) sin A cos B - sin B cos A

Key Concepts and Properties

The key concepts involved include:

The angle sum and difference identities for sine and cosine. The symmetry properties of sine and cosine functions. The unit circle and coordinate geometry in trigonometry.

Understanding these concepts is essential for grasping the derivation and application of the sin(A - B) identity.

Practical Applications

The identity for sin(A - B) has numerous applications in mathematics and real-world scenarios:

Waves and Oscillations: In physics, this identity is used to describe the phase shift between two waves or oscillations. Engineering: In electrical engineering, it is used to analyze alternating current (AC) circuits and phase differences between signals. Calculus: In calculus, this identity is used to simplify trigonometric expressions in integration and differentiation problems.

Let's look at a practical example involving the simplification of a trigonometric expression using the sin(A - B) identity:

Example: Simplify the expression sin(30° - 15°).

Using the identity sin(A - B) sin A cos B - cos A sin B:

sin(30° - 15°) sin 30° cos 15° - cos 30° sin 15°

Substituting the known values:

sin 30° 0.5

cos 15° frac{sqrt{6} sqrt{2}}{4}

cos 30° frac{sqrt{3}}{2}

sin 15° frac{sqrt{6} - sqrt{2}}{4}

Substituting these values:

sin(30° - 15°) 0.5 * frac{sqrt{6} sqrt{2}}{4} - frac{sqrt{3}}{2} * frac{sqrt{6} - sqrt{2}}{4}

After simplifying, we get:

sin(30° - 15°) frac{sqrt{6}}{4}

Conclusion

The identity for the sine of the difference of two angles, sin(A - B), is a powerful tool in trigonometry with wide-ranging applications. By understanding its derivation and practical applications, we can enhance our problem-solving skills in various fields, from physics to engineering and beyond. This identity showcases the beauty and utility of trigonometric relationships, underscoring the importance of mastering these fundamental concepts.