Properties and Relationships

of Trigonometric Ratios

"Understanding the fundamental properties that make trigonometry a powerful tool in mathematics and science"
Welcome to the world of trigonometric ratios! Let's discover the fundamental properties that make these ratios so powerful and universally applicable.
📐 Size Independence

Trigonometric ratios depend only on the angle, not the size of the triangle. Similar triangles with the same angles have identical ratios!

sin θ, cos θ, tan θ
Same for all similar triangles
📊 Range Limitations

For acute angles, sin θ and cos θ are always between 0 and 1, while tan θ can take any positive value.

0 < sin θ ≤ 1, 0 < cos θ ≤ 1
Bounded by the hypotenuse
🔄 Reciprocal Relationships

Each trigonometric ratio has a reciprocal relationship: cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent respectively.

csc θ = 1/sin θ, sec θ = 1/cos θ
Reciprocal relationships
⚡ Pythagorean Identity

The fundamental identity sin²θ + cos²θ = 1 allows us to find all other ratios when one is known, using the Pythagorean theorem.

sin²θ + cos²θ = 1
The fundamental identity