Right Triangles and Trigonometry: Ratios of Right Triangles

Examples of Sine, Cosine and Tangent

Let’s look at a few examples together.

Example 1:
Name the sine, cosine and tangent 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.

sine of A equals the quotient “opposite” divided by “hypotenuse” equals the quotient 12 divided by 12 the square root of 5 
cosine of A equals the quotient “adjacent” over “hypotenuse” equals the quotient 24 divided by 12 the square root of 5 
tangent of A equals the quotient “opposite” over “adjacent” equals twelve-twenty-fourths 

*Note that the final answers should all be rationalized and simplified.

sine of A equals the quotient 12 divided by 12 the square root of 5 equals the quotient 1 divided by the square root of 5 times the quotient the square root of 5 divided by the square root of 5 equals the quotient the square root of 5 divided by 5 
cosine of A equals the quotient 24 divided by 12 the square root of 5 equals the quotient 2 divided by the square root of 5 times the quotient the square root of 5 divided by the square root of 5 equals the quotient 2 the square root of 5 divided by 5
tangent of A equals the quotient “opposite” over “adjacent” equals the quotient 12 divided by 24 equals one-half 

Example 2:
Using the 30°-60°-90° Special Right Triangle, what is the sine, cosine and tangent 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 

sine of 30 degrees 3 equals the quotient “opposite” over “hypotenuse” equals one-half 
c0 sine of 30 degrees 3 equals the quotient “adjacent” over “hypotenuse” equals the quotient the square root of 3 divided by 2
tangent of 30 degrees equals the quotient “opposite” over “adjacent” equals the quotient 1 divided by the square root of 3

*Again, note that the final answers should all be rationalized and simplified.

tangent of 30 degrees equals the quotient 1 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 the square root of 3 divided by 3