Right Triangles and Trigonometry: Ratios of Right Triangles

Examples of Cosecant, Secant and Cotangent

Let’s look at a couple of examples of the reciprocal functions.

Example 1:
Name the cosecant, secant and cotangent of angle A in the figure below.

Right triangle with hypotenuse of length 12 square root of 5 and leg adjacent to angle A with length 24 and leg opposite angle A with length 12.
Right Triangle

 

Solution:
The hypotenuse is 12 the square root of 5 .
The side opposite angle A is 12.
The side adjacent to angle A is 24.

the cosecant of A equals the quotient 1 divided by the sine of A equals the quotient “opposite” over “hypotenuse” equals the quotient 12 the square root of 5 all over 12 
the secant of A equals the quotient 1 divided by the cosine of A equals the quotient “opposite” over “hypotenuse” equals the quotient 12 the square root of 5 all over 24 
the cotangent of A equals the quotient 1 divided by the tangent of A equals the quotient “adjacent” divided by “opposite” equals twenty-four-twelfths 

*Note that final answers should be rationalized and simplified.

the cosecant of A equals the quotient 12 the square root of 5 all over 12 equals the square root of 5 
the secant of A equals the quotient 12 the square root of 5 all over 24 equals the quotient the square root of 5 divided by 2 
the cotangent of A equals twenty-four-twelfths equals 2 

Example 2:
Using the 30°-60°-90° Special Right Triangle, what is the cosecant, secant and cotangent of 30°?

30-60-90 Special Right Triangle
30-60-90 Special Right Triangle

 

Solution:
The hypotenuse is 2.
The side opposite the 30° angle is 1.
The side adjacent to the 30° angle is the square root of 3 .

the cosecant of 30 degrees equals the quotient 1 divided by the sine of 30 degrees equals the quotient “opposite” over “hypotenuse” equals the quotient 2 divided by 1   
the secant of 30 degrees equals the quotient 1 divided by the cosine of 30 degrees equals the quotient “adjacent” over “hypotenuse” equals the quotient 2 divided by the square root of 3 
the cotangent of 30 degrees equals the quotient 1 divided by the tangent of 30 degrees equals the quotient “adjacent” over “opposite” equals the quotient the square root of 3 divided by 1 

*Note again that final answers should be rationalized and simplified.

the secant of 30 degrees equals the quotient 2 divided by the square root of 3 times the quotient the square root of 3 divided by the square root of 3 equals the quotient 2 the square root of 3 divided by 3