Right Triangle Trigonometry    

Section 6.5


A triangle has three sides measured in linear units and three angles measured in degrees or radians whose sum is 180 degrees or p (pi) radians, respectively.

This book only uses degrees for angle measurement.


Recall a right triangle has one angle = 90 degrees, so the sum of the other two must = 90.


Two special triangles:


A Square with sides = x and a diagonal forms two isosceles right triangles.  (An isosceles triangle has two sides equal and the angles opposite them are equal.)

Apply the Pythagorean Theorem to find the length of the diagonal.

        x2 + x2 = (diagonal) 2

        2x2 = (diagonal) 2

        diagonal =


For a right triangle with 45 and 90 degree angles and length of legs, x, the length of the hypotenuse is .


Example 1.  

What is the length of the hypotenuse of a right triangle with legs of length 3 inches each?



Deriving a 30 - 60 - 90 triangle

An equilateral triangle has all sides equal, thus all angles are equal.  Each angle is 60 degrees.

Apply Pythagorean Theorem to find the height.

If each side is 2x in length, then

        (2x) 2  = x2  + (height) 2

          4x2   x2  = (height) 2

          height =  x


An acute angle q , has measurement between  0o < q  < 90o

Since a right angle is 90o, then the other two interior angles of a right triangle must be acute angles.


Trigonometric ratios:

In a right triangle with angle q ,


Sine q   = sin q   =    length of side opposite q / Length of hypotenuse


Cosine q   = cos q   = length of side adjacent q / Length of hypotenuse


Tangent q   = tan q   = length of side opposite q / Length of side adjacent q


Find the exact values using the information obtained for special right triangles:


sin 45o =                                 cos 45o =                                tan 45o

sin 30o =                                 cos 30o =                                tan 30o =

sin 60o =                                 cos 60o =                               tan 60o =



More examples in class.


When triangles are not special 45o or 30-60-90o triangles, use your calculator. 

Set calculators to degree mode!

EX. 2:  Use a calculator to find the following.


a) sin 15o =                                        b) cos 29.5o                                   c) tan 37.2o =

When the angle is not given, but the length of the sides are given, we can find the angle measurement by taking the inverse of the trig functions.

Recall the inverse of f(x) is written f -1(x). 

For a right triangle with hypotenuse of length c, and legs of length b opposite q  and length a adjacent to q :


q = sin-1(b/c)               means     sin q  = b/c


q  = cos-1(a/c)            means     cos q  = a/c


q = tan-1(b/a)             means     tan q  = b/a


Practice using your calculator.


Exercise 3:  


a)  sin-1(1/2) =                        b)  cos-1(2/3) =                                               c)  tan-1(1)



Exercise 4: In class, see diagram.

A building x feet tall has a wire attached to the ground 125 feet away from the base of building to the top of the building.  The angle of elevation of the wire is 33.4o.  Another wire is connected from the same place on the ground to the top of a billboard sitting on top of the building.  The angle of elevation of this wire is 41o.

How tall is the billboard?

(Hint: First find the height of the building.)


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