The Geometric Proof of sin(A B) and Its Misconceptions

Many students encounter the trigonometric identity sin(A B) sin A cos B cos A sin B, but there seems to be a common misconception that sin(A B) sin A sin B. Let's explore this misunderstanding and provide a geometric proof for the true trigonometric identity.

The Misconception

Often, one might encounter statements such as:

“You can’t prove it because it’s not true. Where do you come up with this baloney? Sin AB Sin A x Cos B Cos A x Sin B.”

This statement is incorrect. While it is indeed incorrect to claim that sin(A B) sin A sin B, the statement provided is misleading. The correct identity is sin(A B) sin A cos B cos A sin B. Let's explore why this is the case.

Counterexample

Consider A B π/2. By substituting these values into both expressions, we can see why the claim is false:

sin A B sin(π/2 π/2) sin π 0

sin A * sin B sin(π/2) * sin(π/2) 1 * 1 1

Clearly, sin(π/2 π/2) ≠ sin(π/2) * sin(π/2), which demonstrates the contradiction.

Geometric Proof

Now, let's explore the true identity sin(A B) sin A cos B cos A sin B through a geometric proof. Consider the figure described with points P, O, Q, R, M, and N, where angle QOR A and angle POQ B. PM is perpendicular to OR, PQ is perpendicular to OQ, and QN is perpendicular to PM.

Let's break down the proof step-by-step:

Angle NQO Angle QOR A. Angle PQN 90° - A. Therefore, Angle NPQ A. Triangle OPM is a right-angled triangle. From the definition of sine and cosine in the context of right-angled triangles, we have:

sin(A B) PM/OP

Since NM QR and NMPN is a common segment, we get:

PM NMPN/OP QR/PN * OP QR/OQ * OQ/OP * OP QR * OQ/OQ * OP

Thus, sin(A B) QR/OQ * OP

QR/OQ Cos A, and OQ/OP Sin B, and OP/PQ Sin A, and PQ/OP Cos B.

Therefore, sin(A B) Cos A * Sin B Sin A * Cos B.

This proof clearly demonstrates that the identity sin(A B) sin A cos B cos A sin B holds true, and it’s not equal to sin A sin B.

Conclusion

It's important to understand the difference between sin(A B) and sin A sin B. The former is a trigonometric identity that can be proven geometrically, while the latter is simply a product of sine functions. Understanding these distinctions is crucial for solving trigonometric problems accurately.

Further Reading and Resources

To gain a deeper understanding of trigonometric identities and their applications, consider exploring the following resources:

Books on trigonometry and mathematics Online courses on geometry and trigonometry Mathematical websites and forums for further discussion and practice