1. 10 Trigonometry (1) Contents 10.1 Basic Terminology of Trigonometry
10.2 Definition of Trigonometric Ratio of Any Angles
10.3 Finding Trigonometric Ratios Without Using a Calculator
10.4 Trigonometric Equations
10.5 Graphs of Trigonometric Functions
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10.6 Graphical Solutions of Trigonometric Equations

2. 10.1 Basic Terminology of Trigonometry
A. Angles of Rotation
Fig 10.3 shows a circle on a rectangular coordinate plane, with centre at the origin O and of radius r. A is a point on the circle.
Fig. 10.3
Fig. 10.3
An angle θ is formed by rotating OA about the origin O to reach OP as shown in Fig The angle θ is called the angle of rotation.
We call OA the initial side of the angle and OP the terminal side of the angle.

3. 10.1 Basic Terminology of Trigonometry
B. Quadrants
In a rectangular coordinate plane, the x-axis and y-axis divide the plane into four regions which are known as quadrants. They are labelled and are shown in Fig
Fig. 10.7
The x-axis and y-axis do not belong to any of the four quadrants.

4. 10.1 Basic Terminology of Trigonometry
An angle of rotation θ is said to lie in a quadrant if its terminal side lies in that quadrant.
When the measure of θ is either negative or greater than 360º, the position of the terminal side of θ determines the quadrant in which it lies.
For example, the terminal side of –240º coincides with that of 120º as shown in Fig
Fig. 10.8
Since 120º lies in quadrant II, –240º also lies in quadrant II.

5. 10.2 Definition of Trigonometric Ratio of Any Angles
A. Definition
The trigonometric ratios of an acute angle  are defined as the ratios between the sides of a right-angled triangle.
To define the trigonometric ratios for an arbitrary angle, we can first place triangle OPQ on a rectangular coordinate plane as shown in Fig
The trigonometric ratios of the angle  then can be defined in terms of x, y and r, where
Fig

6. 10.2 Definition of Trigonometric Ratio of Any Angles
For angles greater than 90o, we can consider the terminal side OP of the angle θ and let the coordinates of P be (x, y):
θ lies in quadrant II
θ lies in quadrant III
θ lies in quadrant IV
Fig (a)
Fig (b)
Fig (c)
Then the trigonometric ratios of θ are defined in terms of x, y and r,
as follows:
where

7. 10.2 Definition of Trigonometric Ratio of Any Angles
The following identities hold for all measures of θ.
(a)
(b)
Proof of (a):
By Pythagoras’ Theorem,
Proof of (b):

8. 10.2 Definition of Trigonometric Ratio of Any Angles
B. Signs of the Three Trigonometric Ratios
The signs of the three trigonometric functions can be summarized in Fig This figure is also called a ‘ASTC’ diagram.
Notes:
In the ‘ASTC’ diagram, the letters A, S, T and C indicate the trigonometric ratios that are positive, where
A: All positives;
S: Sin positives;
T: Tan positives; and
C: Cos positives.
Fig

9. 10.3 Finding Trigonometric Ratios Without Using a Calculator
A. Trigonometric Ratios of Angles Formed by the Coordinate Axes
Fig (a)
Fig.10.27(c)
Fig.10.27(b)
Fig.10.27(d)

10. 10.3 Finding Trigonometric Ratios Without Using a Calculator
The values of the trigonometric ratios of ,
and
can be found and the results are listed in Table 10.3. θ
sin θ
cos θ
tan θ 0º 1
90º
undefined
180º –1
270º
360º
As shown in Fig (d), the terminal sides OP of 　　 and lie in the same position. Hence the trigonometric ratios of and are the same.
Table 10.3

11. 10.3 Finding Trigonometric Ratios Without Using a Calculator
B. Trigonometric Ratios by Considering the Reference Angles
The positive acute angle formed between the terminal side of the angle  and the x-axis is called reference angle  as shown in Fig (a) – (d).
 lies in quadrant I
 lies in quadrant III
 lies in quadrant II
 lies in quadrant IV
Fig (a)
Fig (c)
Fig (b)
Fig (d)

12. 10.3 Finding Trigonometric Ratios Without Using a Calculator
C. Finding Trigonometric Ratios when other Trigonometric Ratios are Given
Example 10.3T If
, where ,
find
and .
Solution:
Since
and θ lies in quadrant II, we take
Since θ lies in quadrant II, then P(–2, 3) is a point on the terminal side of θ as shown in the figure.

13. 10.3 Finding Trigonometric Ratios Without Using a Calculator
Consider (–2, 3) to be a point on the terminal side of θ. Then use the definitions to find other trigonometric ratios of .
By definition,

14. 10.4 Trigonometric Equations
A. Inverses of the Trigonometric Ratios
If a trigonometric ratio of an angle is given, then we can find the angle. This process is called finding the inverse of the trigonometric ratio.

15. 10.4 Trigonometric Equations
Example 10.5T If
, where ,
find θ.
Solution:
Since tan θ < 0, θ must lie in either quadrant II or quadrant IV.
Step 1:
Step 2:
Let β be the reference angle of θ. Then
Step 3:
By locating θ and its reference angle diagrammatically as shown in the figures, or

16. 10.4 Trigonometric Equations
B. Algebraic Solutions of Trigonometric Equations
An equation involving trigonometric ratios of an unknown angle is called a trigonometric equation.
For example, (2 sin x – 1)(sin x + 1) = 0 is a trigonometric equation.
It is not an identity because it holds for some values of x only, such as x = 30º. x = 30º is called a solution of this trigonometric equation.

17. 10.4 Trigonometric Equations
Example 10.7T
Solve the equation
for
(Give the answers correct to 1 decimal place.)
Solution:
If cos  = 0, then
sin  =  1 , which does not satisfy 2sin  + cos  = 0.
As cos   0, we can divide the expression by cos  .
(correct to 1 decimal place)

18. 10.5 Graphs of Trigonometric Functions
A. The Graph of y = sin x x 0o
30o
60o
90o
120o
150o
180o
210o
240o
270o
300o
330o
360o y
0.5
0.9 1
–0.5
–0.9 –1
Table 10.4
The ordered pairs in Table 10.4 are plotted and the graph of y = sin x is drawn and shown in Fig
Fig

19. 10.5 Graphs of Trigonometric Functions
Properties of Sine Function:
Fig
Maximum value of sin x = 1 , which corresponds to x = 90o .
Minimum value of sin x = –1 , which corresponds to x = 270o .
The curve of y = sin x for
repeats itself in every 360o.

20. 10.5 Graphs of Trigonometric Functions
B. The Graph of y = cos x x 0o
30o
60o
90o
120o
150o
180o
210o
240o
270o
300o
330o
360o y 1
0.9
0.5
–0.5
–0.9 –1
Table 10.5
Similarly, cos x repeats itself in every 360o. cos x is a periodic function with a period of 360o and
for all values of x.
Fig

21. 10.5 Graphs of Trigonometric Functions
C. The Graph of y = tan x x 0o
30o
60o
75o
90o
105o
120o
150o
180o y
0.6
1.7
3.7
undefined
–3.7
–1.7
–0.6 x
210o
240o
255o
270o
285o
300o
330o
360o y
0.6
1.7
3.7
undefined
–3.7
–1.7
–0.6
Table 10.6
Fig

22. 10.5 Graphs of Trigonometric Functions
Properties of tangent function :
Fig
, the tangent function exhibits the following behaviour:
1. For
From 0o to 90o , tan x increases from 0 to positive infinity.
From 90o to 180o , tan x increases from negative infinity to 0.
2. tan x is a periodic function with a period of 180o.

23. 10.6 Graphical Solutions of Trigonometric Equations
To solve the equation from the graph, draw a straight line as shown below.
The straight line cuts the graph of at P and Q.
The approximate solution is 228 or 312.
To solve the equation from the graph, draw a straight line as shown below.
The straight line does not intersect with the graph of