Basic Concepts of Trigonometry
I. Introduction: Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It has widespread applications in various fields, including physics, engineering, computer science, and more.
Basic Concepts:
I. Angles:

Definition:
 An angle is formed when two rays share a common endpoint, called the vertex.

Measurement:
 Angles are measured in degrees (${}^{\circ}$) or radians (rad).
II. RightAngled Triangle:

Definition:
 A rightangled triangle has one angle that measures exactly 90 degrees.

Sides:
 The side opposite the right angle is the hypotenuse, and the other two sides are the adjacent side and the opposite side.
III. Trigonometric Ratios:

Sine ($\mathrm{sin}\theta $):
 $\mathrm{sin}\theta =\frac{\text{Oppositeside}}{\text{Hypotenuse}}$

Cosine ($\mathrm{cos}\theta $):
 $\mathrm{cos}\theta =\frac{\text{Adjacentside}}{\text{Hypotenuse}}$

Tangent ($\mathrm{tan}\theta $):
 $\mathrm{tan}\theta =\frac{\text{Oppositeside}}{\text{Adjacentside}}$
V. Unit Circle:

Definition:
 The unit circle is a circle with a radius of 1 centered at the origin in a coordinate plane.

Coordinates:
 For an angle $\theta $ in standard position, the coordinates of the point where the terminal side intersects the unit circle are $(\mathrm{cos}\theta ,\mathrm{sin}\theta )$.
VI. Trigonometric Identities:

Pythagorean Identity:
 ${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$

Reciprocal Identities:
 $\mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta}$
 $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta}$
 $\mathrm{cot}\theta =\frac{1}{\mathrm{tan}\theta}$

Quotient and CoFunction Identities:
 $\mathrm{tan}\theta =\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}$
 $\mathrm{cot}\theta =\frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}$
VII. Trigonometric Functions:

Periodicity:
 Sine and cosine functions have a period of $2\pi $or 360 degrees.

Graphs:
 The graphs of sine and cosine functions oscillate between 1 and 1.

Amplitude and Frequency:
 The amplitude is the maximum value of the function, and the frequency is the number of cycles per unit.