# 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:

1. Definition:

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

• Angles are measured in degrees (${}^{\circ }$) or radians (rad).

II. Right-Angled Triangle:

1. Definition:

• A right-angled triangle has one angle that measures exactly 90 degrees.
2. 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:

1. Sine ($\mathrm{sin}\theta$):

2. Cosine ($\mathrm{cos}\theta$):

3. Tangent ($\mathrm{tan}\theta$):

V. Unit Circle:

1. Definition:

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

• For an angle $\theta$ in standard position, the coordinates of the point where the terminal side intersects the unit circle are $\left(\mathrm{cos}\theta ,\mathrm{sin}\theta \right)$.

VI. Trigonometric Identities:

1. Pythagorean Identity:

• ${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$
2. Reciprocal Identities:

• $\mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta }$
• $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$
• $\mathrm{cot}\theta =\frac{1}{\mathrm{tan}\theta }$
3. Quotient and Co-Function Identities:

• $\mathrm{tan}\theta =\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }$
• $\mathrm{cot}\theta =\frac{\mathrm{cos}\theta }{\mathrm{sin}\theta }$

VII. Trigonometric Functions:

1. Periodicity:

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

• The graphs of sine and cosine functions oscillate between -1 and 1.
3. Amplitude and Frequency:

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