How to Solve cos30°tan30° Using Trigonometric Identities and Triangle Properties

Introduction

Understanding and solving trigonometric expressions like cos30°tan30° is fundamental in various fields, including mathematics, physics, and engineering. This article provides a step-by-step guide to solving this expression, using both trigonometric identities and geometric properties.

Step-by-Step Breakdown

Using Trigonometric Identities

To solve cos30°tan30°, we can use the tan θ sin θ / cos θ identity. Here's how:

Recall the identity tan θ sin θ / cos θ. Substitute tan 30° with sin 30° / cos 30° in the expression:

cos30°tan30° cos30° * (sin30° / cos30°)

Notice that cos30° in the numerator and denominator cancel out, leaving:

sin30°

We know that:

sin30° 1/2

Therefore:

cos30°tan30° 1/2

Using Right Triangle Properties

Alternatively, we can use the properties of a 30-60-90 right triangle to determine the values of cos30° and tan30°.

A 30°-60°-90° triangle is an equilateral triangle with a side of 2, split into two congruent right triangles by an altitude. By using the Pythagorean theorem, we can determine the length of the sides:

The hypotenuse is 2, one leg (opposite the 30° angle) is 1, and the other leg (adjacent to the 30° angle) is sqrt3.

Using the definitions of trigonometric functions:

cos30° adjacent / hypotenuse sqrt3 / 2 tan30° opposite / adjacent 1 / sqrt3

Now, we can multiply these values:

cos30°tan30° (sqrt3 / 2) * (1 / sqrt3) 1 / 2

Conclusion

By using either trigonometric identities or geometric properties, we have shown that cos30°tan30° 1/2. Understanding these methods is crucial for solving similar trigonometric expressions and is applicable in various mathematical and real-world scenarios.