11
◙ EP-Program
- Strisuksa School - Roi-et
Math
: Trigonometry (3)
► Dr.Wattana Toutip - Department of Mathematics – Khon Kaen University
© 2010 :Wattana Toutip
◙ [email protected]
◙ http://home.kku.ac.th/wattou
11. Radians
Often angles are measured in terms of radians rather than degree.
If a sector of a circle with radius r subtends an arc of length  , then the angle of the sector
measured in radians is  / r .(Fig 11.1)
1
There are 2 radians in a circle. A right angle is  radians.
2
Fig 11.1
Fig 11.2
To convert degree to radians, multiply by 
.
180
To convert radians to degrees, multiply by 180  .
Suppose a circle has radians r . Then a sector which subtends  radians at the centre of the
1
circle has area r 2 (Fig 11.2)
2
The length of an arc which subtends  radians at the centre of a circle radius r is r .
Angular velocity is measured in radians per second.
Inverse trig .functions
The ranges of the inverse trig. functions are as follows.
1
1
For sin and cosec .      
2
2
For cos and sec . 0    
1
1
For tan and cot .      
2
2
11.1.1 Examples
1. Solve the equation tan x  3 ,giving all the solutions in range 0 to 2 .
Solution
If you have a scientific calculator, then it will directly in radians.
Fig11.3
If your calculator works only in degrees, then
multiply tan 1 3 by  180 .In both case the answer is
tan 1 3  1.25 .
There is another solution between  and 3 2 .
x  1.25 or   1.25  4.39
2.
A circle with radius 10 cm is cut by a
chord which subtends 0.8 radius at the centre .(Fig 11.3).
Find the area of the smaller segment.
Solution
1
The area of the sector is 102  0.8 .
2
1
The area of the triangle is 102  sin 0.8 .
2
The area of the segment the difference between these .
1
Area  102 (0.8  sin 0.8)  4.13cm2
2
11.1.2 Exercises
1. Convert the following to radians , leaving your answers in terms of  .
(a) 180
(b) 60
(c) 240
(d) 30
(e) 10
2. Convert the following to degree.
1
(a) 
3
(b) 1.3
(c) 3.7
(d) 4
1
(e) 
4
3. A wheel is spinning at 50 revolutions per minute. What is its angular speed?
4. Convert an angular speed of 20 radians per second to revolutions per minute.
5. What are the angular speeds of the minute and hour hands of a clock ?
6. A wheel of radius 2 m rolls along the floor at an angular speed of 35 .What is the
speed of the wheel in m s 1 ?
7. The wheel of a cycle is of radius 45 cm. What is the angular speed of the wheel
when the cycle is moving at 4 m s1 ?
8. What is the angular speed of
(a) The earth about its axis
(b) The earth about the sun?
9. A sector of a circle radius 5 cm has angle 0.4 . Find the arc length of the sector
and the area of the sector .
10. An arc of length 3 cm lies on a circle of radius 15 cm. What angle dose the arc
subtend at the centre?
11. An arc of length 5 cm subtends an angle of 1.1. What is the radius if circle ?
12. A chord subtends 1.05 at the centre of a circle radius 12 cm. Find the areas of the
minor and major segments.
13. Two circle, each of radius 10 cm, have their centre 15 cm apart. Find the area
which is common to both the circles.
14. A circle of radius r has a sector of which the arc length is a . Find the area of the
sector in terms of r and a .
15. Solve the following equations, giving the solutions in the range 0 to 2 .
(a) sin x  0.2
(b) cos x  0.4
(c) tan x  2
1
(d) cot x  
2
2
(e) sec x  tan x  1
5
(f) cos ecx  tan x
2
1

(g) sin    x   0.1
3

1

(h) tan    x   1.8
2

(i) 6sin 2 x  5sin x  1  0
16. Use your calculator to verify that the inverse functions sin1 , cos 1 , tan 1 do lie
within the ranges given.
17. Without a calculator find the following
 1
(a) sin 1   
 2
(b) cos 1  1
(c) cot 1  1
(d) sin 1  sin 2 

 1 
(e) cos 1  cos     
 6 

(f) tan 1  tan 2 
18. Show that for 1  x  1 , cos 1 x  sin 1 x is constant, and find its value.
11.2 Small angles
If  is a small angle measured in radians, then the following approximations hold .
1
sin    , tan    ,cos   1   2
2
11.2.1
Example
 sin 
Find an approximation for
,valid when  is small.
1  cos 
Solution
Rewrite the function using the approximations above.
 sin 
 

1  cos 
 1 
1  1   2 
 2 

2
1 2

2
 sin 
2
1  cos 
11.2.2
Exercises
1. Find approximations for the following, valid when  is small.
sin 
(a)
tan 
sin 2
(b)

(c)
1  cos 
2
sec   1
(d)
2
sin  tan 
(e)
1  cos 3
1

(f) tan     
4

1

(g) cot    2 
4

1
1

tan       tan 
4
4

2. Find an expression for
, valid when  is small.

1
1
1
1
3. Use the values sin   , cos  
3 to find an approximation for
6
2
6
2
1

sin      .
6

1
1

cos       cos
3
3

4. Find an approximation for
.

5. Show that sin1 

. Find a small value of x which is an approximate solution
180
of sin x  cos x  1.05 .
11.3 Examination questions
Fig 11.4
1. The diagram shows two arcs, AB and CD , of concentric circles, centre O .The
radii OA and OC are 11cm and 14 cm respectively
, and AOB   radian . Express in terms of  the
area of
(i)
Sector AOB
(ii)
The shaded region ABCD .
Given that the area of shaded region ABCD
is 30cm3 , calculate
(iii) The value of 
(iv)
The perimeter of the shaded
region ABCD .
2. After t seconds the extension of a piston is given by x  sin 3t and of its
crankshaft by x  sin 2t .
(a) Sketch these two curves on the same axes for 0  t  2 .
(b) Prove that sin  t  2t   3sin t  4sin 3 t .
(c) Hence verify that the expression of the piston is the same as that of the
crankshaft when S  S  1 4S  3  0 , where S  sin t .
(d) Calculate, corrected to 3 significant figures, all the times in the first five
seconds when the expressions are the same.
(e) Hence find the percentage of the time in each cycle for which the piston is
more extended than the crankshaft.
3. State the exact principal values, in radians, of
(i) sin 1  1
1
(ii) cos 1  
2
(iii) sin 1  0.4   cos 1  0.4  .
[Warning. Because your calculator works to a finite number of figure it may
not give you the exact values.]
4.
(a) (i) Express 7sin   24cos  in the form R sin     , giving the value
1
of R and  , where 0     .
2
(ii) What is the minimum value of 7sin   24cos  , and for what value of
x between 0 and 2 does it occur?
1
(iii)What is the minimum value of
for values of 
7sin   24cos   5
between  2 and 3 2 , and for what value of  does it occur?
(b) (i) A sector of a circle, centre O , has radius r and A . If the length of the
1
arc of the sector is s , prove that A  rs .
2
Fig 11.5
(ii) In the diagram, A and B are points on a circle,
centre O , such that the length of the arc AB  x .The
tangent BC of length x is drawn and OC is joined,
meeting the circle at D . Prove that the shaded area is
equal to the area of the sector OAD .
5.
Use the approximations sin x  x;
1
tan x  x;cos x  1  x 2 to find the smallest positive root
2
of the equation sin x  cos x  tan x  1.5 giving your
answer to 2 decimal places.
Common errors
1. Radians
When working with radians, make sure that your calculator is correctly adjusted.
If a question is about radians, then it will tell you directly , or it will ask you to
give the answer in the range 0 to 2 . You will get very few marks if you then
give your answer in degrees.
2. Small angles
The approximations for sin, cos and tan of small angles hold only when the angle
is measured in radians. They do not apply if the angle is given in degrees.
Solution (to exercise)
11.1.2
1.
(a) 

(b)
3
4
(c)
3

(d)
6

(e)
18
2.
(a) 60
(b) 74.5
(c) 212
(d) 710
(e) 45
3. 5.24
4. 191
5. 1.75 103 ,1.45 104
6. 70 m/s
7. 8.9
8.
(a) 7.27 105
(b) 1.99 107
9. 2cm , 5cm 2
1
10.
2
11. 4.55cm
12. 13.1, 439
13. 45.3
1
14. ra
2
15.
(a) 0.201, 2.940
(b) 1.982, 4.301
(c) 1.107, 4.249
(d) 2.034,5.176
3 7
,
4 4
(f) 0.685, 2.457,3.826,5.598
(e) 0,  ,
(g) 1.994,5.336
(h) 0.507,3.649
(i)
1
 , 6.12,3.31
2
17.
(a) 
(b) 
(c) 
(d) 0

6

4

6
(f) 2  
(e)
18.
1

2
11.2.2
1.
(a) 1
(b) 2
1
(c)
2
1
(d)
2
4
(e)
9
1
(f)
1
1  2
(g)
1  2
2. 2
1 1
3.

3
2 2
1
4. 
3
2
1  
5. 1   
2  90 
6. 0.45
7. 0.051
2
===========================================================
References:
Solomon, R.C. (1997), A Level: Mathematics (4th Edition) , Great Britain, Hillman
Printers(Frome) Ltd.
More: (in Thai)
http://home.kku.ac.th/wattou/service/m123/12.pdf
http://home.kku.ac.th/wattou/service/m456/08.pdf