Trigonometry

Trigonometry

Introduction

Trigonometry is a vital branch of mathematics that studies the relationships between the angles and sides of triangles. Understanding trigonometric functions is essential for various fields, including engineering, physics, and geometry.

Key Trigonometric Functions

The fundamental trigonometric functions include sine, cosine, and tangent. These functions are defined based on a right triangle or the unit circle.

Definitions:

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

tan⁡(θ)=oppositeadjacent=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}

Key Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any angle. Here are some essential identities:

  • Pythagorean Identities:
    • sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1
    • 1+tan⁡2(x)=sec⁡2(x)1 + \tan^2(x) = \sec^2(x)
    • 1+cot⁡2(x)=csc⁡2(x)1 + \cot^2(x) = \csc^2(x)
  • Angle Addition Formulas:
    • sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b)\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
    • cos⁡(a+b)=cos⁡(a)cos⁡(b)−sin⁡(a)sin⁡(b)\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)
    • tan⁡(a+b)=tan⁡(a)+tan⁡(b)1−tan⁡(a)tan⁡(b)\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane. It is a vital tool for understanding trigonometric functions and the angles associated with them.

Unit Circle

Conclusion

Trigonometry connects angles and lengths, helping to solve problems in various fields. By mastering the core functions and identities, students can apply these concepts to real-world scenarios and advanced mathematical studies.