1. Trigonometric Ratios and
their Graphs

2. The Trigonometric Ratios for any angle90
180
360
270
-90
-180
-270
-360 90
180
270
-90
-180
-270
-360
360
180o 0o
450o 0o
90o
180o
270o
360o
-90o
-180o
-270o
-360o
-450o
270o 1
sinx + circle 0o
90o
180o
270o
360o  -1

3. Sine, cosine and tangent for any angle
The trigonometric ratios have previously been used to solve problems in right-angled triangles only. We need to develop an approach that will solve problems involving triangles with angles greater than 900.
SOHCATOA O A H 
Sin CosTan ?
In order to be able to do this we have to define sine, cosine and tangent in a different way. For angles greater than 90o we can establish a connection between the trigonometric ratios and the moving point on the circumference of a circle of unit radius. O 1

4. Sine, cosine and tangent for any angle
As the Point P moves in an anti-clockwise direction around the circumference of the circle, the angle  changes from 0o to 360o. O 1 P
(x,y) A y 
Consider the right-angled triangle formed by the vertical line PA. x x
In this triangle the distance OA = x.
The distance OP = y. x O 1 P  A y
So point P has co-ordinates (x,y).
(cos ,sin )
sin 
cos 
Therefore x = cos  and y = sin .
So the co-ordinates of P are (cos , sin ).

5. Sine, cosine and tangent for any angle
We are now in a position to define the sine and cosine of any angle including angles greater than 90o
cos 
sin 
(cos ,sin ) x O 1 P  A y x y
We can now state the values of the sine and cosine for any angle. Consider the co-ordinates of the point P as it moves around the circle through the angles shown below.
(0.5,0.87)
(-0.5,0.87) P
120o P
60o P
245o P
310o
(-0.42,-0.91)
(0.64,-0.77)
cos 60o = 0.5 sin 60 = 0.87
cos 120o = sin 120 = 0.87
cos 245o = sin 245 =
cos 310o = sin 310 =

6. Use your calculator to plot the graph of y = sin x on the grid below.
-0.87
0.5
1.0
-1.0
0.0
-0.87
0.5
0.87
0.87
0.0
-0.5
-0.5
0.00 1
y = sin x
The sine curve is symmetrical above the x axis about 90o and below the x axis about 270o
0.8 7 8 9 4 5 6 1 2 3
Sin 
0.6
0.4
0.2 x 90
180
270
360
- 0.2
- 0.4
sin 300 = sin 150o
sin 600 = sin 120o
- 0.6
In general sin x =sin(180 - x)
- 0.8
- 1

7. Use your calculator to plot the graph of y = cos x on the grid below.1
y = cos x
0.8
0.6
0.4
0.2
0.0
1.0
-0.5
1.00
-0.87
-1.0
0.87
-0.87
-0.5
0.5
0.5
0.87
0.0 x 90
270 7 8 9
180
360
- 0.2 4 5 6
- 0.4 1 2 3
- 0.6
Cos 
- 0.8
- 1

8. Use your calculator to plot the graph of y = cos x on the grid below.1
y = cos x
0.8
0.6
0.4
cos 300 = - cos 150o
0.2 x 90
180
270
360
- 0.2
- 0.4
In general cos x = - cos(180o - x)
- 0.6
- 0.8
cos 600 = - cos 120o
- 1

9. 90
180 x
270
360
-90
-180
-270
-360 1 -1
y = Sin x O
-60o
y = Cos x
The graphs of the sine and cosine functions are also defined for angles that are negative. This corresponds to the point P going around the circle in a clockwise direction.

10. 90
180 x
270
360
-90
-180
-270
-360 1 -1
y = Sin x
-315o
-225o
135o O
-60o
If sin 45o = 0.71, use the graph above to find three more angles whose sine also has this value.
If cos 68o = 0.37, use the graph below to estimate three more angles whose cosine also has this value. 90
180 x
270
360
-90
-180
-270
-360 1 -1
y = Cos x
-292o
-68o
292o

11. 3Sinx 2Sinx Sinx Period 360o 3 Amplitude  3 Period 360o 21
Amplitude  1 x
-360
-270
-180
-90 90
180
270
360 -1 -2 -3

12. 3Cosx 2Cosx ½Cosx Cosx 90 180 x y = f(x) 270 360 -90 -180 -270 -360 1
270
360
-90
-180
-270
-360 1 -1 2 -2 3 -3
3Cosx
2Cosx
½Cosx
Cosx

13. Some relationships between the sine and cosine ratios in different quadrantsy y 90
180
360
270
-90
-180
-270
-360
150o
50o
30o
30o x x
-50o
Sin (-50o)= -Sin 50o
Sin(180o - 30o) = Sin 150o = Sin 30o
Sin (-) = -Sin 
Sin (180 - ) = Sin  y y 90
180
270
-90
-180
-270
-360
360
150o
50o x
30o
30o x
-50o
Cos (-50o)= Cos 50o
Cos (180o - 30o) = Cos 150o = -Cos30o
Cos (180 - ) = -Cos 
Cos (-) = Cos 

14. The Tangent Ratio for Angles greater than 90 DegreesP Q R 
Tan  can be expressed in terms of the sine and cosine functions. This relationship can be determined by considering the three ratios in a right-angled triangle.
Can you use the ratios above to write tan  in terms of sin  and cos ?
Clue 1
Clue 2
Clue 3
This is true for all angles of .

15. S+ C- S+ C+ T- T+ S- C- T+ T- S- C+ y x
60o
30o
90o
120o
150o
180o
210o
240o
270o
300o
330o
360o
The sine and cosine functions can have values that are either positive or negative depending on the size of . It is useful to identify the quadrants in which their values are positive or negative. S+ C- S+ C+
Quadrant 1 2 3 4 T- T+
What about the tangent ratio? S- C- T+ T- S- C+

16. S A C T Science All Teachers Care Sin+ ALL+ Cos+ Tan+ S+ C- S+ C+ T-x y
60o
30o
90o
120o
150o
180o
210o
240o
270o
300o
330o
360o
It is very useful to be able to recollect all this information. One way is just to remember the quadrants in which the ratios are positive. There is a well known mnemonic to help you remember this. Whether you agree with it or not is another matter!
Sin+
ALL+
All S+ C- S+ C+
Science S A
Quadrant 2
Quadrant 1 T- T+
Quadrant 3
Quadrant 4 C T
Teachers S- C- T+ T- S- C+
Care
Tan+
Cos+

17. Origin of the Tangent Function1 
cos 
sin 
Some Tangents!

18. Origin of the Tangent Function
Some Tangents! 1 
cos 
sin 
This is the reason that this ratio is called the tangent.
tan 

19. tan    and is not defined when cos  = 01
cos 
sin  
tan    and is not defined when cos  = 0
Cos 90o = 0
sin+
cos-
tan-
sin+
cos+
tan+ A S T C
sin-
cos-
tan+
sin-
cos+
tan-
Cos 270o = 0

20. Obtuse and reflex angles can be written in terms of an acute angle.
1. Sin 145o
2. cos 250o
3. tan 300o
4. sin 220o
5. cos 330o
6. tan 210o
We will write each of the following in terms of an acute angle with the aid of diagrams and by considering symmetry.
145o
250o
70o
300o
220o
330o
210o
35o
40o
30o
35o
70o
60o
40o
30o
30o
sin 35o
-cos 70o
-tan 60o
-sin 40o
cos 30o
tan 30o
1.In the 2nd quadrant sin is positive.
2. In the 3rd quadrant cos is negative.
3. In the 4th quadrant tan is negative.
4.In the 3rd quadrant sin is negative.
5. In the 4th quadrant cos is positive.
6. In the 3rd quadrant tan is positive.

21. The Graph of the Tangent Function
y = tan  0o
90o
180o
270o
360o
-90o
-180o
-270o
-360o
-450o
450o
Tan  is not defined when cos  = 0.
That is, for angles of 90o , 270o , 450o ….
and -90o , -270o , -450o ….

22. Solving Equations Involving the Trigonometric Functions
Example 1: Find all solutions to 2 sinx = 1 in the range -360o to 360o.
Step 1
Solve the equation for x using your calculator if necessary.
2sinx = 1
So the solutions are
30o , 150o , - 210o and - 330o.
 sinx = 0.5
 x = 30o
Step 2
(Graph not given)
Picture the quadrants in your mind’s eye or make a rough sketch and go from there, considering the symmetry of the situation and the positive and negative movement of the point.
30o
150o
-210o
-330o

23. Solving Equations Involving the Trigonometric Functions
Example 1: Find all solutions to 2 sinx = 1 in the range -360o to 360o.
Step 1
Solve the equation for x using your calculator if necessary.
2sinx = 1
So the solutions are
30o , 150o , - 210o and - 330o.
 sinx = 0.5
 x = 30o
Step 2
(Graph given)
Draw line y = 1 and locate the first solution (x = 30o). Then by considering the symmetry of the situation, read off all other solutions at the intersections with the graph. 90
180
270
360
-90
-180
-270
-360
y = 2Sin x 1 2 -1 -2

24. Solving Equations Involving the Trigonometric Functions
Question 1: Find all solutions to 4cosx = 3 in the range -360o to 360o.
Step 1
Solve the equation for x using your calculator if necessary.
4cosx = 3
So the solutions are
41.4o , 318.6o ,
- 41.4o and o.
 cosx = 0.75
 x = 41.4o
Step 2
(Graph not given)
Picture the quadrants in your mind’s eye or make a rough sketch and go from there, considering the symmetry of the situation and the positive and negative movement of the point.
41.4o
318.6o
-41.4o
-318.6o

25. Solving Equations Involving the Trigonometric Functions
Question 1: Find all solutions to 4cosx = 3 in the range -360o to 360o.
Step 1
Solve the equation for x using your calculator if necessary.
4cosx = 3
 cosx = 0.75
 x = 41.4o
Step 2
(Graph given)
So the solutions are
41.4o , 318.6o ,
- 41.4o and o.
Draw line y = 3 and locate the first solution (x = 41.4o). Then by considering the symmetry of the situation, read off all other solutions at the intersections with the graph. 90
180
270
360
-90
-180
-270
-360
y = 4Cos x 1 2 3 4 -1 -2 -3 -4

26. Solving Equations Involving the Trigonometric Functions
Example 1: Use the graph below to solve sin3x = sin 60o for all values of x in the range 0o to 270o.
Step 1
Solve the equation for x using your calculator if necessary.
sin3x = sin 60o
 x = 60o
 x = 20o (sin 60o = 0.87)
Step 2
Draw the line y = 0.87 on the graph and read off the solutions at the points of intersection. 1 -1 60
120
180
240
300
y = Sin 3x
From the graph solutions are: 20o, 40o, 140o, 160o and 260o.

27. Solving Equations Involving the Trigonometric Functions
Example 1: Use the graph below to solve sin3x = sin 60o for all values of x in the range 0o to 240o.
Step 1
Solve the equation for x using your calculator if necessary.
sin3x = sin 45o
 x = 45o
 x = 15o (sin 45o = 0.71)
Step 2
Draw the line y = 0.71 on the graph and read off the solutions at the points of intersection. 1 -1 60
120
180
240
y = Sin 3x
From the graph solutions are: 15o, 45o, 135o and 165o.

28. x y
60o
30o
90o
120o
150o
180o
210o
240o
270o
300o
330o
360o A S T C
By considering the movement of the point on the above diagrams you should be able to deduce the following identities:
sin (360 + x) = sin x
cos (360 + x) = cos x
tan(360 + x) = tan x
sin (180 - x) = sin x
cos (180 - x) = -cos x
tan(180 - x) = -tan x
sin (- x) = -sin x
cos (- x) = cos x
tan (- x) = -tan x

29. The following diagrams show the relationships between the three trigonometric ratios for a circle of radius 1 unit. Tangent means “to touch”.
Radius P O Q
Half Chord PM
Sin  = O/H = PM/1 = PM
Chord
Cos  = A/H = OM/1 = OM M
Angular Bisector o
Tan  = PT/1 = PT
Tangent T o P O 1 M P M O 1  P O T  1
Sinus
Cosinus M O P 1 

30. Worksheets Various 1 - 1 360 x 90 180 270 0.2 0.4 0.6 0.8 - 0.2 - 0.4
0.2
0.4
0.6
0.8
- 0.2
- 0.4
- 0.6
- 0.8
Worksheets Various

31. 90
180 x
270
360
-90
-180
-270
-360 1 -1 90
180 x
270
360
-90
-180
-270
-360 1 -1

32. 90
180 x
y = f(x)
270
360
-90
-180
-270
-360 1 -1 2 -2 3 -3

33. 90
180
270
360
-90
-180
-270
-360
y = 2Sin x 1 2 -1 -2 90
180
270
360
-90
-180
-270
-360
y = 4Cos x 1 2 3 4 -1 -2 -3 -4

34. 1 -1 60
120
180
240
300
y = Sin 3x 1 -1 60
120
180
240
300
y = Sin 3x