# Trigonometric Ratios

I. Introduction: Trigonometric ratios are fundamental relationships between the angles and sides of a right-angled triangle. They provide a framework for understanding the geometric properties of triangles and have widespread applications in mathematics and various scientific disciplines.

II. Trigonometric Ratios in a Right-Angled Triangle:

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

• Definition:
• Relationship: $\mathrm{sin}\theta =\frac{a}{c}$
2. Cosine ($\mathrm{cos}\theta$):

• Definition:
• Relationship: $\mathrm{cos}\theta =\frac{b}{c}$
3. Tangent ($\mathrm{tan}\theta$):

• Definition:
• Relationship: $\mathrm{tan}\theta =\frac{a}{b}$

III. Reciprocal Trigonometric Ratios:

1. Cosecant ($\mathrm{csc}\theta$):

• Definition:
2. Secant ($\mathrm{sec}\theta$):

• Definition:
3. Cotangent ($\mathrm{cot}\theta$):

• Definition:

IV. Unit Circle Representation:

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

V. Trigonometric Ratios and Angles:

1. Acute Angles:

• Trigonometric ratios are primarily defined for acute angles in a right-angled triangle.
2. Cofunctions:

• The cofunction identities express the relationships between a trigonometric ratio and its cofunction:
• $\mathrm{sin}\left(\frac{\pi }{2}-\theta \right)=\mathrm{cos}\theta$
• $\mathrm{cos}\left(\frac{\pi }{2}-\theta \right)=\mathrm{sin}\theta$
• $\mathrm{tan}\left(\frac{\pi }{2}-\theta \right)=\mathrm{cot}\theta$
• $\mathrm{cot}\left(\frac{\pi }{2}-\theta \right)=\mathrm{tan}\theta$
• $\mathrm{sec}\left(\frac{\pi }{2}-\theta \right)=\mathrm{csc}\theta$
• $\mathrm{csc}\left(\frac{\pi }{2}-\theta \right)=\mathrm{sec}\theta$