Pythagorean Theorem

One of the most famous mathematicians who has ever lived, Pythagoras, a Greek scholar who lived way back in the 6th century B.C. (back when Bob Dole was learning geometry), came up with one of the most famous theorems ever, the Pythagorean Theorem.  It says - in a right triangle, the square of the measure of the hypotenuse equals the sum of the squares of the measures of the two legs.  This theorem is normally represented by the following equation: a2 + b2 = c2, where c represents the hypotenuse.

With this theorem, if you are given the measures of two sides of a triangle, you can easily find the measure of the other side.



1. Problem: Find the value of c.


 



 
Solution: a2 + b2 = c2 Write the Pythagorean

Theorem and then plug in any

given information.


 
52 + 122 = c2 The information that was

given in the figure was

plugged in.


 

 
169 = c2 Solve for c

c = 13




 


One of the special right triangles which we deal with in geometry is an isosceles right triangle.  These triangles are also known as 45-45-90 triangles (so named because of the measures of their angles).  There is one theorem that applies to these triangles.  It is stated below.

In a 45-45-90 triangle, the measure of the hypotenuse is equal to the measure of a leg multiplied by SQRT(2).

The following figure presents the theorem in graphical terms.






 


There's another kind of special right triangle which we deal with all the time.  These triangles are known as 30-60-90 triangles (so named because of the measures of their angles).  There is one theorem that applies to these triangles.  It is stated below.

In a 30-60-90 triangle, the measure of the hypotenuse is two times that of the leg opposite the 30o angle.  The measure of the other leg is SQRT(3) times that of the leg opposite the 30o angle.

The following figure presents the theorem in graphical terms.






While the word trigonometry strikes fear into the hearts of many, we made it through (amazing as it may seem to us), and hope to help you through it, too!  Each of the three basic trigonometric ratios are shown below.



sine of angle A = (measure of opposite leg)/(measure of hypotenuse).  In the figure, the sin of angle A = (a/c).

cosine of angle A = (measure of adjacent leg)/(measure of hypotenuse).  In the figure, the cos of angle A = (b/c).

tangent of angle A = (measure of opposite leg)/(measure of adjacent leg).  In the figure, the tan of angle A = (a/b).



1. Problem: Find sin A, cos A, and tan A.


  
Solution: sine = (opposite/hypotenuse)   

sine = 5/13


 

 
cosine = (adjacent/hypotenuse)

cos = 12/13


 
tangent = (opposite/adjacent)

tan = 5/12

Be aware that, although the example above seems to indicate otherwise, the values for the trigonometric ratios depend on the measure of the angle, not the measures of the triangle's sides.


Many problems ask that you find the measure of an angle or a segment that cannot easily be measured.  Problems of this kind can often be solved by the application of trigonometry.  Below is an example problem of this type.

1. Problem: A ladder 12 meters long leans

against a building. It rests on

the wall at a point 10 meters

above the ground. Find the angle

the ladder makes with the ground.


 
Solution: Make sure you know what is being

asked. Then use the given

information to draw and label a

figure. Here's our idea of a

figure for this problem:


  
Choose a variable to represent the

measure of the angle you are asked

to find. Using the variable you

have chosen, write an equation that

will solve the problem.


 
sin x2 = (10/12)


 
The above equation is derived from

the given information and the

knowledge of the sine

ratio.


 
Find the solution using a calculator's

Arcsine function or a table

of trigonometric ratios.


 
TI-82 screen: sin-1 (10/12) = 56.44


 
Trigonometric Ratios Table:

sin 56o = 0.8290

sin 57o = 0.8387


 
By either answer, after rounding to

the nearest degree, the answer is 56o.