Evaluating sinX cosX at Xπ/3 Using Trigonometric Identities

Trigonometry is a fundamental branch of mathematics that explores the relationships between the angles and the sides of triangles. In this article, we will delve into the problem of evaluating (sin(x) cos(x)) when (x frac{pi}{3}), using various trigonometric identities, properties of triangles, and conversions between degrees and radians. This evaluation will be crucial in understanding the behavior and relationships within trigonometric functions.

Introduction to Trigonometric Identities

Trigonometric identities are essential for simplifying and solving various trigonometric equations. One such identity that will be particularly useful here is (2 sin(x) cos(x) sin(2x)). This identity allows us to convert the product of sine and cosine functions into a single sine function, making the evaluation process more straightforward.

Evaluating (sin(x) cos(x)) at (x frac{pi}{3})

Given the expression (T sin(x) cos(x)), we start by using the identity mentioned earlier:

(2 sin(x) cos(x) sin(2x))

Substituting this into the original expression, we obtain:

(sin(x) cos(x) frac{1}{2} sin(2x))

Now, let's evaluate this at (x frac{pi}{3}):

(sin(frac{pi}{3}) cos(frac{pi}{3}) frac{1}{2} sin(2 cdot frac{pi}{3}) frac{1}{2} sin(frac{2pi}{3}))

To further simplify, we can express (frac{2pi}{3}) in terms of degrees since (pi ) radians equals 180 degrees:

(frac{2pi}{3} frac{2 cdot 180}{3} 120°)

Using the value of (sin(120°)), which can be evaluated as:

(sin(120°) sin(180° - 60°) sin(60°) frac{sqrt{3}}{2})

Substituting back, we get:

(sin(frac{2pi}{3}) frac{sqrt{3}}{2})

Hence,

(sin(frac{pi}{3}) cos(frac{pi}{3}) frac{1}{2} cdot frac{sqrt{3}}{2} frac{sqrt{3}}{4})

Using Triangle Properties

Another approach involves using the properties of a special type of triangle: an equilateral triangle. An equilateral triangle has all angles equal to (60°), and since the sum of angles in a triangle is (180°), we have (3 cdot 60° 180°). When constructing a perpendicular from one vertex to the opposite side, we obtain two right-angled triangles.

In the right-angled triangle with angles 60°, 90°, and 30°, we know:

(sin(60°) frac{sqrt{3}}{2}) and (cos(60°) frac{1}{2})

Thus,

(sin(frac{pi}{3}) cos(frac{pi}{3}) frac{sqrt{3}}{2} cdot frac{1}{2} frac{sqrt{3}}{4})

Additional Insights

We can also derive the values of (sin(frac{pi}{3})) and (cos(frac{pi}{3})) using the properties of a 30-60-90 triangle. An equilateral triangle with side lengths of 1 can be split to form a 30-60-90 triangle.

In this triangle:

(sin(frac{pi}{3}) frac{sqrt{3}}{2}) and (cos(frac{pi}{3}) frac{1}{2})

This is consistent with our earlier results, confirming that: