Trigonometric Functions

Trigonometric functions describe the relationship between the sides and angles of a right triangle. They play a vital role in many areas of mathematics, physics, and engineering.

Sine, Cosine, and Tangent

The three most common trigonometric functions are sine, cosine, and tangent. They are defined as follows:

• Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side.

These functions can be used to find the missing sides and angles of right triangles.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions. They can be used to solve problems involving angles and triangles.

Some common types of trigonometric equations include:

• Identity equations: Equations that are true for all angles, such as sin²θ + cos²θ = 1.
• Periodicity equations: Equations that involve the period of a trigonometric function, such as sin(θ + 2π) = sinθ.
• Pythagorean identity: The equation sin²θ + cos²θ = 1, which is used to find the missing sides of a right triangle.

Trigonometric Applications

Trigonometric functions have numerous applications in various fields:

• Astronomy: To calculate the positions and motion of celestial objects.
• Navigation: To determine the location of ships and airplanes.
• Surveying: To measure distances and angles.
• Architecture: To design buildings and structures.
• Physics: To describe the motion of objects and waves.

Trigonometry is a fundamental tool for understanding the world around us and solving problems in a wide range of applications.