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What Trigonometric Ratio Can Be Represented by $\sin\theta \cos\theta$?

Published in Trigonometric Identities 3 mins read

The trigonometric expression $\sin\theta \cos\theta$ can be exactly represented by half of the sine of the double angle, specifically as $\frac{1}{2}\sin(2\theta)$. This relationship is derived from one of the fundamental double angle identities in trigonometry.

Understanding the Double Angle Identity

The core of this representation lies in the double angle identity for sine, which states:
$\sin(2\theta) = 2 \sin\theta \cos\theta$

To express $\sin\theta \cos\theta$ in terms of $\sin(2\theta)$, we simply rearrange this identity:
$\sin\theta \cos\theta = \frac{1}{2}\sin(2\theta)$

This identity allows us to transform a product of sine and cosine functions of a particular angle into a single sine function of double that angle, scaled by a constant factor of one-half.

Foundations of Trigonometric Ratios

To better understand the components of the expression $\sin\theta \cos\theta$, let's revisit the definitions of sine and cosine in a right-angled triangle:

  • Sine (sin θ): This ratio is defined as the length of the side opposite to angle θ divided by the length of the hypotenuse.
  • Cosine (cos θ): This ratio is defined as the length of the side adjacent to angle θ divided by the length of the hypotenuse.

The expression $\sin\theta \cos\theta$ thus combines these fundamental ratios through multiplication, leading to a new form that can be simplified using the double angle identity.

Practical Applications and Insights

The ability to represent $\sin\theta \cos\theta$ as $\frac{1}{2}\sin(2\theta)$ is incredibly useful across various mathematical and scientific disciplines:

  • Simplification of Expressions: It provides a method to simplify complex trigonometric expressions, making them easier to analyze or integrate.
  • Calculus: In integral calculus, this identity is frequently used to simplify integrals involving products of sine and cosine, converting them into a simpler form that is easier to integrate. For example, $\int \sin x \cos x \, dx$ can be transformed into $\int \frac{1}{2}\sin(2x) \, dx$.
  • Solving Trigonometric Equations: This identity can help in solving trigonometric equations by transforming a product of functions into a single function of a related angle, which can often be more manageable.
  • Physics and Engineering: In fields like wave mechanics, signal processing, and electrical engineering, such transformations are crucial for simplifying models that describe oscillatory phenomena.

Example: Calculating $\sin\theta \cos\theta$ for a Specific Angle

Let's illustrate with an example where $\theta = 30^\circ$:

  1. Direct Calculation:

    • $\sin 30^\circ = \frac{1}{2}$
    • $\cos 30^\circ = \frac{\sqrt{3}}{2}$
    • $\sin 30^\circ \cos 30^\circ = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}$
  2. Using the Identity:

    • $\frac{1}{2}\sin(2\theta) = \frac{1}{2}\sin(2 \times 30^\circ) = \frac{1}{2}\sin(60^\circ)$
    • Since $\sin 60^\circ = \frac{\sqrt{3}}{2}$,
    • $\frac{1}{2}\sin(60^\circ) = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}$

Both methods yield the same result, confirming the accuracy and utility of the identity.