Question

Which expression is equivalent to sin(1.8x) sin(0.5x)?

Answer 1

The expression sin(1.8x) sin(0.5x) is equivalent to ½[cos(1.3x) − cos(2.3x)] using the product-to-sum trigonometric identity.

Step-by-step explanation:

The expression sin(1.8x) sin(0.5x) can be rewritten using a trigonometric identity known as the product-to-sum identity. Specifically, we will use the identity for the product of sines, which states that sin A sin B is equivalent to ½[cos(A − B) − cos(A + B)]. Applying this to our expression, we get:

sin(1.8x) sin(0.5x) = ½[cos(1.8x − 0.5x) − cos(1.8x + 0.5x)]

By simplifying the terms inside the cosine functions:

sin(1.8x) sin(0.5x) = ½[cos(1.3x) − cos(2.3x)]

This simplification results in an expression that combines two cosine waves with different frequencies. The product-to-sum formula is particularly useful in physics and engineering contexts, such as analyzing wave interference or signal processing.

Answer 2

Hum, this problem was difficult. You use the next expression to solve this problem. \[\cos (A - B) = \cos A \cos B + \sin A \sin B \] \[\cos (A + B) = \cos A \cos B - \sin A \sin B\] \[\cos (A - B ) - \cos (A +B ) =2 \sin A \sin B\] So \[\sin A \sin B = 0.5 \left( \cos(A - B) - \cos(A + B) \right)\] A = 1.8 x, B = 0.5 x \[\sin (1.8x) \sin (0.5x) = 0.5\left( \cos(1.8-0.5)x - \cos(1.8+0.5)x \right)\]\[= 0.5 \left( \cos(1.3x) - \cos (2.3x) \right)\] It's finish !!