S2 - G4 : Pythagoras' theorem and Trigonometry

Pythagoras' Theorem

For a right angle triangle:-

If the hypotenuse has length c, and the sides have lengths a and b, then

c² = a² + b²  

[hypotenuse = the length across or opposite the right angle / longest length]

=> if the sides of a triangle have lengths a, b and c, such that c² = a² + b², 

      then the triangle is a right-angle triangle.

Example 

What is c?

Using c² = a² + b²

                  5² + 12²

                =  25 + 144

                = √169

                = 13

Example         

If every square is 1 unit, what is a?

 Using c² = a² + b² 

           25 = 5², 9 = 3²  

           5² = a² + 3²   

           a²  = 25 - 9

           a²  = √16
             a = 4

Trigonometric Ratios

  Tangent ,Cosine and Sine

    To find angle x or any of the sides, we can use   

               Tan x  = Opposite   =  b                 TOA

                             Adjacent       a

                Cos x =   Adjacent  = a                 CAH     

                             Hypotenuse   c

               

                Sin x =    Opposite   = b                SOH

                            Hypotenuse    c

Example

Find 

     (a) length AC

     (b) Sin X

     (c) Cos X

     (d) Tan X

     (e) Angle X

a. Length AC

      c² = a² + b²            Step 1 : which formula to use? Label a=6, b = 8

                                                  [Right angle ◺, use Pythagoras Theorem]

      c² = 6² + 8²            Step 2  : Compute and answer

      c = √36 + 64  

         = 10

    AC = 10 cm

b. Sin X =  Opp/Hyp     Step 1 : Determine the formula and label 

             = 6/10              Step 2 : Compute and answer  

             = 3/5              

c.  Cos X = Adj/Hyp      Step 1 : Determine the formula and label

             = 8/10              Step 2  : Compute and answer

             = 4/5               

d.  Tan X = Opp/Adj

               = 6/8 = 3/4

e. Tqn X = 6/8               Step 1 : Determine the formula and label

          X = Tan^-1(6/8)      Step 2  : Compute and answer

             =  36.86^o

[If possible, always use the value given to do the computation instead of a computed value. This is to avoid wrong answer in case there is of a mistake in the computed value]

Angle of Elevation and Depression

The angle is measured between the horizontal line (AB) and the line forming the angle. 

    

            Angle of Elevation – Looking Up

            Angle of Depression – Looking Down

Sine Rule

            a       =     b     =    c

          SinA        SinB      SinC                 

                         or 

          SinA  =  SinB   =   SinC

             a            b             c

Cosine Rule

             a² = b² + c² – 2bcCosA

Note:

1. The value of Sinθ and Cosθ is between -1 to 1 =>

                    -1  ≤ Sinθ ≤ 1

                    -1  ≤ Sinθ ≤ 1

    Check your computation if your answer for Sinθ orCosθ is not within -1 to 1

2.  For Pythagoras Theorem, the Hypotenuse length is the longest 

     => the value of both sides are smaller than the hypotenuse length.

     Check your computation  if any side has a bigger value than the hypotenuse. 

3.  If possible, always use the given values in the question for computation.

Practice

Find 

     (a) length BC

     (b) Sin Y

     (c) Cos Y

     (d) Tan Y

     (e) Angle Y

     (f)  Sin Z

     (g) Cos Z

     (h) Tan Z

     (i)  Angle Z

Answer

a. AC² = AB² + BC²        Step 1 : Right Angle ◺, use Pythagoras T]

      13² = 5² + BC²         Step 2 : Label Diagram and Compute

      BC² = 13² - 5²

      BC = √169 - 25

            = √144 

            = 12

b.  Sin Y = O/H               Step 1 : Determine the formula and labe

               = 12 / 13          Step 2 : Label Diagram and Compute

c.  Cos Y = A/H

               = 5/13

d.  Tan Y = O/A

               = 12/5

e.   Y = Cos ^-1 (5/13)

         = 67.38^o   

f.   Sin Z  = 12/13

g.  Cos Z = 5/13

h.  Tan Z = 5/12

i. Angle Z = 180 - 90 - 67.38^o     

                = 22.62^o   

[If possible, always use the given values in the question; this is to avoid subsequent wrong answers if there is error in a computed value.]