Trigonometry is the study of the relationships between the angles and sides of triangles, primarily focused on right-angled triangles.

'Trigonometry' comes from Greek words 'trigon', meaning triangle, and 'metron', meaning measure.

An angle is defined by the rotation of a ray around a vertex, measured from the initial side to the terminal side.

Positive angles are measured anticlockwise, while negative angles are measured clockwise from the initial side.

A radian is a unit of angular measure defined as the angle subtended by an arc equal in length to the radius of the circle.

An angle is measured in degrees by dividing one complete revolution (360 degrees) into 360 equal parts.

The main trigonometric functions are sine (sin), cosine (cos), tangent (tan) along with their reciprocals: cosecant (csc), secant (sec), and cotangent (cot).

For an angle θ in a right triangle, sin(θ) = opposite side / hypotenuse.

For an angle θ in a right triangle, cos(θ) = adjacent side / hypotenuse.

For an angle θ in a right triangle, tan(θ) = opposite side / adjacent side.

The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ.

Confusing the ratios; for instance, treating cosine as sine or vice versa.

The identity sin(θ) = cos(90° - θ) is fundamental for complementary angles.

The unit circle, with a radius of 1, provides values for trigonometric functions at key angles (e.g., 0°, 30°, 45°, 60°, 90°).

Trigonometry is used in navigation, physics, engineering, and many fields like architecture and astronomy.

The graphs show the values of sine, cosine, and tangent as functions of the angle, displaying periodic behavior.

The range of sin(θ) and cos(θ) is [-1, 1].

Secant (sec) is the reciprocal of cosine, and cosecant (csc) is the reciprocal of sine.