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Trigonometry is a branch of mathematics that deals with the study of relationships involving angles and lengths of triangles. It has wide applications in various fields such as physics, engineering, astronomy, and architecture.

Basic Concepts

Trigonometry primarily focuses on the trigonometric functions, which include sine ($$\sin$$), cosine ($$\cos$$), tangent ($$\tan$$), cosecant ($$\csc$$), secant ($$\sec$$), and cotangent ($$\cot$$). These functions relate the angles of a triangle to the lengths of its sides.

Key Relationships

Some of the key relationships in trigonometry include:

• The Pythagorean theorem, which relates the lengths of the sides of a right triangle:
  $$a^2+b^2=c^2$$,
  
• The definitions of trigonometric functions in terms of the sides of a right triangle:
  $$\sin \left(\theta \right)=\frac{opposite}{hypotenuse}$$,
  $$\cos \left(\theta \right)=\frac{adjacent}{hypotenuse}$$,
  $$\tan \left(\theta \right)=\frac{opposite}{adjacent}$$,
  
• The unit circle, which provides a way to define trigonometric functions for all real numbers using the coordinates of points on the circle.

Applications

Trigonometry is used in various real-world applications, including:

• Navigation and surveying
• Mechanical engineering and construction
• Astronomy and celestial navigation
• Signal processing and telecommunications
• Computer graphics and animation

Understanding trigonometry is essential for solving problems involving angles and triangles in many fields of science and engineering.