Trigonometric Graphs

Trigonometric Graphs

Remember the trigonometric ratios and quadrants?

The trigonometric angles can also be represented in a graphic form.

A sample Trigonometric Graph ( y = sin x)

1. The graph has a maximum and minimum point.

2. One complete cycle takes 360^o.

Y = Sin X

1.  The graph has a complete cycle of 360^o.

2. The graph is above 0 in the first and second quadrants(0^o - 180^o)

       => SinX is positive, the value is greater than 0.

3. The graph is below 0 in the third and fourth quadrants (180^o - 360^o)

       => SinX is negative, the value is less than 0.

4. The maximum value of SinX is 1.

5.  The minimum value of SinX is -1.

y = Cos X

1.  The graph has a complete cycle of 360^o.

2. The graph is above 0 in the first and 4th quadrants(0^o - 90^o, 270^o - 360^o)

       => SinX is positive, the value is greater than 0.

3. The graph is below 0 in the second and third quadrants (180^o - 270^o)

       => SinX is negative, the value is less than 0.

4. The maximum value of CosX is 1.

5.  The minimum value of CosX is -1.

Y =  Tan X

1.  The graph has a complete cycle of 360^o

2. The graph is above 0 in the first and third quadrants(0^o - 90^o, 270^o - 360^o)

       => TanX is positive, the value is greater than 0.

3. The graph is below 0 in the second and fourth quadrants (180^o - 270^o)

       => TnX is negative, the value is less than 0

4. There is no maximum or minimum value as both go towards to infinity

Trigonometry Ratio, Angles/Radian and Quadrants

 Circle and Quadrants

So far, we have worked on the trigonometric value of acute angles. Let's look at angles beyond 90^o

Computation of X      

   2nd Quadrant : X = 180^o - y.   (where y = x in the first quadrant)

   3rd Quadrant : X = 180^o + y

   4th Quadrant = 360^o - y

For angles between 360^o and 720^o,

   1st Quadrant : X - 360^o

   2nd Quadrant : X = (180+360)^o - y  (where y = x in the first quadrant 360^o-540^o)

   3rd Quadrant : X = (180+360)^o - y 

   4th Quadrant :  X =  (360 + 360)^o - y

Angles beyond 720^o are calculated similarly

Degree and Radian (Unit of Measures) for angles
An angle can also be measured using radian as a unit of measure.
=> Both degree and radian is a measure of angle.

Radian is defined as the angle from the centre of a circle which intercepts an arc equal in length to the radius of the circle.

Both degree and radian is a measure of angle.

                  360^o = 2π rad

                 => 1^o = 2π rad / 360^o

  

         2π rad = 360^o

           =>   1 rad = 360^o / 2π

Converting between Degree and Radian

                  360^o = 2π rad

Example

What is the value of 240^o in rad?

    360^o = 2π rad

  240^o = 2π rad x 240^o

                 360^o

           = 4π/3 rad

Trigonometric Angles and Quadrants

SinX, CosX and TanX

First Quadrant : 0^o - 90^o 

All values of SinX, CosX and TanX are positive values.

Second Quadrant : 90^o - 180^o 

SinX has a positive value.

CosX and TanX have negative values.

Third Quadrant : 180^o - 270^o 

TanX has a positive value.

CosX and SinX have negative values.

Fourth Quadrant : 270^o - 360^o 

CosX has a positive value.

TanX and SinX have negative values.

The positive values for SinX, TanX and CosX can be remembered by:

         Quadrant:      1st     2nd      3rd    4th      

                               All Students Take Coffee

                                     I              A       O

                                    N             N       S

S3N3 Obtuse Angle - SinX and CosX, Sine and Cosine Rule

Extending Sin X and Cos X to Obtuse Angle

     Sin A = Sin (180 - A)

     Cos A = -Cos (180 - A)

Sine Formula

Area of Triangle

Area of  ⃤ ABC 

  = 1/2 ab Sin C 

  = 1/2 bc SinA 

  = 1/2 ac SinB

Area =  √ s(s -a) (s-b) (s - c) where 2s = a + b + c

Sine Rule

            a       =     b     =    c

          SinA        SinB      SinC                 

                         or 

          SinA  =  SinB   =   SinC

             a            b             c

Cosine Rule

             a² = b² + c² – 2bcCosA

             Cos A = b² + c² - a² / 2bc

              Cos B = a² + c² - b² / 2ac

              Cos C = a² + b² - c² / 2ab

Bearing

When reading a compass bearing, the directions are from either north of the south. 

From the North, 060^o is N60^oE

From the South, 225^o is S45^oW

Decimal values are to be included in all compass reading. 

S3A3: Trigonometric function Y = A sin(Bx + C) + D

Trigonometric Function Y = A Sin (Bx + C) + D

A trigonometric function can be of the form:

   Y = A Sin (Bx + C) + D or Y = A Cos (Bx + C) + D 

   (A, B, C and D are numbers)

A : Amplitude => the height of the graph , A is always positive => |A| modulus 

B : Period => Number of Cycles within 360^o  or 2π

C : Phase Shift : C/B => Moving the graph Left/Right along the x-axis

D : Vertical Shift => Moving the graph Up/Down along the y-axis

A : Amplitude=> Height of the graph

Example

Draw the graph Y = 2SinX

=> A = 2, B = 1, C = 0 , D = 0, A => Height from centre to the maximum/minimum point

=> Y = 2 Sin (1x + 0) + 0

=> y = 2 Sin X

   

 |A|  = Max value - Min value / 2

  Y = 2sinX : The “height” of the graph increased by 2 times from the centre

B : Period => Number of Cycles within 360^o  

Y = Sin2X

=> A =1, B = 2, C = 0 , D = 0

=> Y = 1 Sin (2x + 0) + 0

=> y = Sin 2X

To find the number of cycles within 360^o

Period = 2π / B   or     360^o/B

            = 2π / 2           360^o / 2

             = π                  180^o

=> The entire cycle is within π, 180o

From the graph, there are 2 cycles within 360^o

=> B indicates the number of cycles within 360^o

      i.e. 2 => 2 cycles, 3 => 3 cycles , etc

C : Phase Shift : C/B => Moving the graph Left/Right

Y = Sin (X + 60^o)

=> A = 1, B = 1, C = 60^o , D = 0

=> Y = 1Sin (1x + 60^o) + 0

=> y =  Sin( X + 60^o)

Positive C “shifts” the graph to the left.

Y = Sin (X - 60^o)

=> A = 1, B = 1, C = -60^o , D = 0

=> Y = 1Sin (1x + 60^o) + 0

=> y =  Sin( X - 60^o)

Negative C “shifts” the graph to the Right.

D : Vertical Shift => Moving the graph Up/Down

Y = SinX + 1

=> A = 1, B = 1, C = 0 , D = 1

=> Y = 1Sin(X + 0) + 0

=> y =  SinX + 1

Positive D “shifts” the graph upward along the y-axis.

Y = SinX - 1

=> A = 1, B = 1, C = 0 , D = 1

=> Y = 1Sin(X + 0) + 0

=> y =  SinX + 1

Negative D “shifts” the graph downward along the y-axis.

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.]

S1G1 : Angles

TYPES OF ANGLES

           

PROPERTIES OF ANGLES

         

        ㄥ at a point = 360^o       ㄥon a straight line=180^o          Sum ㄥ of triangle = 180^o

                                                                                                 a + b+ c = 180

                                                                                       b + d = 180(ㄥ on str line)

      

Vertically opposite angles           Alternate Angle         Corresponding Angles

ㄥA = ㄥC,                                           ㄥa = ㄥd                         ㄥA = ㄥE, ㄥC=ㄥG, 

ㄥD = ㄥB                                            ㄥc = ㄥd                        ㄥB = ㄥF , ㄥD = ㄥH 

Complimentary angles
For a right angle, two or more angles add up to 90^o 

NA/O Level

Triangles

                             

            Perimeter = adding the three side 

                    Area = ½ x Base  x Height


                        

                       Area of triangle ABC = ½ ab sinC 

Circles
                                 
                      Angle at centre is twice angle at circumference