Sin, Cos, Tan Formulas
Before diving into the sin, cos, and tan formulas, let’s recall what these terms represent. Sin, cos, and tan are three of the six primary trigonometric ratios used to solve problems involving right triangles. The other three major trigonometric ratios are cot, sec, and cosec. This article will explore the sin, cos, and tan formulas in detail.
What Are Sin, Cos, and Tan Formulas?
The sin, cos, and tan formulas are defined for a right triangle. Consider a right triangle  with a right angle at . Here,  is the hypotenuse (the longest side),  is adjacent to angle , and  is opposite to angle . The trigonometric ratios for angle  are as follows:
Sin, Cos, Tan Formulas
The primary sin, cos, and tan formulas for angle  are:
- sin  = Opposite side / Hypotenuse =Â
- cos  = Adjacent side / Hypotenuse =Â
- tan  = Opposite side / Adjacent side =Â
Reciprocal Relationships
Using the definitions of sin, cos, and tan, we can derive their reciprocal relationships with cosec (or csc), sec, and cot:
- sin  =  (or) csc  =Â
- cos  =  (or) sec  =Â
- tan  =  (or) cot  =Â
Expressing Tan and Cot in Terms of Sin and Cos
The functions tan and cot can also be expressed in terms of sin and cos:
- tan  =Â
- cot  =Â
Using the definitions:
- tan  =  =  =Â
- cot  =  =  =Â
Thus, the formulas that connect tan and cot with sin and cos are:
- tan  =Â
- cot  =Â
Examples Using Sin, Cos, and Tan Formulas
Example 1
Given a right triangle , find the values of sin , cos , and tan .
Let the lengths of the sides be:
- Â = 10 units (hypotenuse)
-  = 6 units (adjacent to )
-  = 8 units (opposite to )
Using the sin, cos, and tan formulas:
- sin  =  =  = 0.8
- cos  =  =  = 0.6
- tan  =  =  = 1.33
By understanding and applying these formulas, you can solve various trigonometric problems involving right triangles.
