# Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides. These functions play a fundamental role in geometry, physics, engineering, and various mathematical applications. Here are key concepts related to trigonometric functions:

1. Basic Definitions:

• Sine Function ($\mathrm{sin}$):

• Defined as the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
• $\mathrm{sin}\left(\theta \right)=\frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine Function ($\mathrm{cos}$):

• Defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
• $\mathrm{cos}\left(\theta \right)=\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent Function ($\mathrm{tan}$):

• Defined as the ratio of the length of the side opposite the angle to the length of the adjacent side.
• $\mathrm{tan}\left(\theta \right)=\frac{\text{Opposite}}{\text{Adjacent}}$
• Cosecant Function ($\mathrm{csc}$), Secant Function ($\mathrm{sec}$), Cotangent Function ($\mathrm{cot}$):

• Reciprocal functions of sine, cosine, and tangent, respectively.
• $\mathrm{csc}\left(\theta \right)=\frac{1}{\mathrm{sin}\left(\theta \right)}$
• $\mathrm{sec}\left(\theta \right)=\frac{1}{\mathrm{cos}\left(\theta \right)}$
• $\mathrm{cot}\left(\theta \right)=\frac{1}{\mathrm{tan}\left(\theta \right)}$

2. Unit Circle:

• Trigonometric functions can be defined on the unit circle, where the radius is 1.
• The coordinates of a point on the unit circle correspond to the values of $\mathrm{sin}$ and $\mathrm{cos}$ for a specific angle.

3. Periodicity:

• All trigonometric functions are periodic with a period of $36{0}^{\circ }$ or $2\pi$ radians.
• $\mathrm{sin}\left(\theta +36{0}^{\circ }\right)=\mathrm{sin}\left(\theta \right)$ and $\mathrm{cos}\left(\theta +2\pi \right)=\mathrm{cos}\left(\theta \right)$

4. Trigonometric Identities:

• Pythagorean Identity:
• ${\mathrm{sin}}^{2}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)=1$
• Reciprocal Identities:
• ${\mathrm{csc}}^{2}\left(\theta \right)={\mathrm{sin}}^{2}\left(\theta \right)+1$
• ${\mathrm{sec}}^{2}\left(\theta \right)={\mathrm{cos}}^{2}\left(\theta \right)+1$
• ${\mathrm{cot}}^{2}\left(\theta \right)=1+{\mathrm{tan}}^{2}\left(\theta \right)$
• Quotient Identities:
• $\mathrm{tan}\left(\theta \right)=\frac{\mathrm{sin}\left(\theta \right)}{\mathrm{cos}\left(\theta \right)}$

5. Trigonometric Formulas:

• Sum and Difference Formulas:
• $\mathrm{sin}\left(\alpha ±\beta \right)=\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right)±\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• $\mathrm{cos}\left(\alpha ±\beta \right)=\mathrm{cos}\left(\alpha \right)\mathrm{cos}\left(\beta \right)\mp \mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• Double-Angle Formulas:
• $\mathrm{sin}\left(2\theta \right)=2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)$
• $\mathrm{cos}\left(2\theta \right)={\mathrm{cos}}^{2}\left(\theta \right)-{\mathrm{sin}}^{2}\left(\theta \right)$
• Half-Angle Formulas:
• $\mathrm{sin}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1-\mathrm{cos}\left(\theta \right)}{2}}$
• $\mathrm{cos}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{2}}$