ep-program - strisuksa school - roi-et 10 math

10. Trigonometric equations

10.1 The six trigonometric ratios

The right –angled triangle is labeled is shown. The six

ratios between the sides are :

sin ;cos ; tanOPP ADJ OPP

HYP HYP ADJ

cot ;sec ;cosADJ HYP HYP

ecOPP ADJ OPP

Some relations between these functions are as follows:

sin cos 1 1

tan ;cot ;sec ;coscos sin cos sin

ec

Pythagoras’s Theorem gives rise to the three identities: 2 2cos sin 1 2 2sec 1 tan

2 2cos 1 cotec

( Here 2cos means 2(cos ) etc ).

10.1.1. Example

Prove the identity: 1

tan cotsin cos

Solution

Write the left hand side in terms of sin and cos , and add. 2 2sin cos sin cos

tan cotcos sin sin cos

Use the identity: 2 2cos sin 1

1tan cot

sin cos

◙ EP-Program - Strisuksa School - Roi-et

Math : Trigonometry (2) ► Dr.Wattana Toutip - Department of Mathematics – Khon Kaen University

© 2010 :Wattana Toutip ◙ [email protected] ◙ http://home.kku.ac.th/wattou

10

mailto:[email protected]

◙ EP .Program – Strisuksa School Roi-et. Mathematics 10. Trigonometric Functions page 2

10.1.2. Exercise

1. Prove the following identities.

(a) 2 2 2cos sin 2cos 1

(b) 4 4 2 2cos sin cos sin

(c) 4 4 2 2sec tan sec tan

(d) 2 2 2 2sec cos sec secec co

(e) 2

2

2

tan 12sin 1

tan 1

(f) 21 12sec

1 sin 1 sin

(g) cos

tan sec1 sin

2. If 3

sin5

x find, without using a calculator , the values of

cos ,cos , tanecx x x .

3. If tan 2x find , in square root form, the values of sec ,cos ,sinx x x .


4. If 2

2sin

1

tx

t

find cos x and tan x in terms of t .

5. Simplify the following expressions:

(a) 21 cos x

(b) 22 tan 2x

(c) 21 sin

cot

x

x

(d) 2

cot

1 cot

x

x

6. If 2cosx and 2siny show that 2 2 4x y .

7. Find an equation involving x and y , without , from the following .

(a) 3cos , 3sinx y

(b) 2cos , 3sinx y

(c) tan , secx y

(d) 1

cot , 2 sec2

x y co

(e) 1 cos , 2 sinx y

(f) 3 2cos , 2 3sinx y

(g) cos sin , cos sinx y

(h) sec tan , sec tanx y .

10.2 Trigonometric functions for other angles

The definition of the trigonometric functions can be extended to angles other

than acute angles .

For 90 180

sin sin(180 ),cos cos 180 , tan tan 180

For180 270

sin sin 180 ,cos cos 180 , tan tan 180


For 270 360

sin sin 360 ,cos 360 , tan tan 360 .

cos ,secec and cot obey the same rule as sin,cos and tan respectively.

For angles greater than 360or less than 0 , sin sin 360 .Similar rules

hold for all the other functions.

10.2.1 Example

1. Solve for 0 180 , cos 2 0.25

Solution

Using the inv cos button, 2 76 .By the formulae above , there

is also a value in the range 270 to 360 .

2 76 or 360 76 284

38 or 142

2. Solve for 20 360 ,2cos sin 1 0 .

Solution

Write 2cos as 21 sin and rearrange. 21 sin 2sin 0 .

This factorize , giving : 1 sin 2sin 1

The first bracket gives sin 1 , so 90 .

The second bracket gives 1

sin2

, which has solution in the

ranges 180 to 270 and 270 to 360 .

90 or 210 or 330

10.2.2 Exercises

1. Express the following in terms of rations of acute angles .

(a) sin156

(b) cos 220

(c) tan 325

(d) cot130

(e) sec370

(f) cos 784ec

2. Solve the following equations, giving all the solutions in the range

0 to 360 .

(a) 1

sin 0.32

x

(b) tan 2 3x

(c) sin 60 0.3x

(d) cos 20 0.4x

(e) sec 4x


(f) cos 50 1.5ec x

(g) 1

cot 2.52

x

(h) 1

cos 0.32

x

(i) sin 30 0.1x

(j) tan 50 1.9x

3. Find x in the range 0 360x , if 3

sin5

x and 4

cos5

x .

4. Find x in the range 0 360x , if 5

cos13

x and 12

sin13

x .

5. If 8

sin17

x find cos x , given that x is between 90 and 180 .

6. If 7

tan24

x find cosecx , given that x is between 90 and 180 .

7. Solve the following equations , giving all the solutions in the

solutions in the range 0 to 180

(a) 2sin 0.5x

(b) 2tan 3x

(c) 3cos secx x

(d) tan 2cotx x

(e) 5cos sinx x

(f) sin tanx x

(g) 2tan 5sec 7 0x x

(h) 4cos sec 4x x

(i) tan cot 2x x

10.3 Compound angles

Trigonometric functions of sums and differences of angles can be found by the following

formulae .

sin sin cos cos sinA B A B A B

sin sin cos cos sinA B A B A B


cos cos cos sin sinA B A B A B

cos cos cos sin sinA B A B A B

tan tan

tan1 tan tan

A BA B

A B

tan tan

tan1 tan tan

A BA B

A B

Important case occur when A B .

sin 2 2sin cosA A A

2 2 2cos cos sinA A A

22cos 1A

21 2sin A

2

2 tantan 2

1 tan

AA

A

The following are known as factor formulae:

1 1

sin sin 2sin cos2 2

C D C D C D

1 1

sin sin 2sin cos2 2

C D C D C D

1 1

cos cos 2cos cos2 2

C D C D C D

1 1

cos cos 2sin sin2 2

C D C D D C

10.3.1 Examples

1. If and are acute angles, for which 3

sin5

and 5

sin13

, find sin

without using a calculator.]

Solution


Using the identity 2 2 4cos sin 1,cos

5A A and

12cos

13 .Apply the formula

above:

sin sin cos cos sin

3 12 4 5

5 13 5 13

56

sin65

2. Use the fact that 1

cos30 32

to find an expression for cos15 , without using a

calculator.

Solution

Let 15A in the formula for cos 2A above. 2cos30 2cos 15 1 .

Make cos15 the subject of this formula.

1 1cos15 3

2 4

3. Prove the identity :

tansin 2 sin 2

sin 2 sin 2 tan

x yx y

x y x y

Solution

Factorize both top and bottom of the left hand side :

1 12sin 2 2 cos 2 2

sin 2 sin 2 2 21 1sin 2 sin 2

2cos 2 2 sin 2 22 2

x y x yx y

x yx y x y

sin cos

cos( )sin( )

x y x y

x y x y

Use the fact that tan sin / cos .

tansin 2 sin 2

sin 2 sin 2 tan

x yx y

x y x y

10.3.2 Exercises

1. Simplify the following expression, without using a calculator .

(a) sin10 cos30 cos10 sin30

(b) cos30 cos50 sin30 sin50

(c) sin cos 2 cos sin 2A A A A

(d) 2 2cos 2 sin 2A A


2. Simplify

(a) sin 40 cos 40

(b) 2

2 tan 20

1 tan 20

3. Letting be the angle such that 3 4

sin ,cos5 5

, express in terms of and

.

(a) 4 3

sin cos5 5

(b) 4 3

cos sin5 5

4. 1 1 3

cos 45 sin 45 ,sin 30 cos60 ,sin 60 cos302 22

.

Use these values to obtain square root expressions for the following .Do nor use a

calculator .

(a) sin 75

(b) sin15

(c) cos75

(d) tan15

(e) cos 22.5

(f) tan 22.5

5. Without using a calculator find the value of tan x , given that tan 45 3x

6. If and are acute angles so that 3

sin5

and 5

sin13

, find the values of

the following.

(a) sin

(b) sin

(c) cos

(d) tan

7. Solve the following equations , for values of between 0 and 180 .

(a) sin 2 1.7sin

(b) cos 2 cos 1

(c) sin 30 cos 60

(d) 4sin 45 cos 45

(e) tan 2 3tan

(f) tan 45 3tan

8. Prove the following identities.

(a) sin( ) sin 2sin cosA B A B A B


(b) cos cos 2cos cosA B A B A B

9. Use the factor formulae to prove the following identities:

(a) cos cos 1

tansin sin 2

A BB A

A B

(b) sin sin 1

tancos cos 2

X YX Y

X Y

(c) sin sin 120 sin 240 0x x x

10. Use the factor formulae to solve the following equations, for 0 180x :

(a) sin sin 60 1x x

(b) cos cos3 0x x

(c) sin5 sin cos2x x

(d) 1

cos 10 cos 102

x x

(e) 1

sin 30 sin 302

x x

(f) cos cos3 sin sin3x x x x

11. Writing 3 2A a A , find an expression for cos3A in terms of cos A only.

12. Find an expression for sin 3A in terms of sin A only.

10.4 The form sin cosa b

2

2 2sin cos sina b a b , where tanb

a .

2

2 2sin cos sina b a b , where tana

b .

10.4.1 Examples

1. Find the range of the function 8sin 15cosy x x .

Solution

Use the formula above.

2 28 15 sin 17siny x x

Whatever the value of , sin x lies between 1 and 1.

17 17y

2. Solve the equation 3sin 4cos 1x x ,for 0 360x .

Solution

Use the formula above to rewrite the equation.

5sin 1x , where 1 4tan 53.1

3 .

1 1sin 11.5

5x or 168.5 or 371.5 .

When is subtracted the first value becomes out of range.

115x or 318


10.4.2 Exercises

1. Write the following in the form sinr x , stating the values of r and .

(a) 3sin 4cosx x

(b) 4sin 5cosx x

(c) 6cos 3sinx x

(d) sin cosx x

(e) 1 1

sin cos2 4

x x

(f) 3 1

sin cos4 8

x x

2. Find the greatest value of the functions in Question 1, and the value of x

which gives the maximum.

3. Write down the range of values of the function in Question 1.

4. Solve the following equations, foe values of x between 180 and 180 .

(a) 2sin 3cos 1x x

(b) 4sin 5cos 3x x

(c) 7sin 8cos 9x x

(d) 12sin 11cos 13x x

5. Write 3sin 4cosy x x in the form sin( )r x .Show that y can also be

written as cos( )s x ,giving the values of s and . Solve the equation

2.5y in the range 180 180x , using both forms of y . Check that

your answer are the same.

10.5 Examination questions

1. Prove the identity:

1 sin cos2sec

cos 1 sin

x xx

x x

2. Solve the equations below, giving all the answer in the range 0 360x :

(i) cos2 0.35x

(ii) tan 4cot 3x x

(iii) 2 28sin 8sin 3cos 0x x x

3. (i) On the same diagram, sketch the graphs of 3cosy x and

2sin 2y x in the interval 0 360x .

(ii) By using the formula for sin 2 3cosx x in the same interval.


(iii)Solve the inequality 2sin 2 3cosx x in this interval.

4. (i) Without the use tables or a calculator , find the values of

(a) sin 40 cos10 cos 40 sin10

(b) 1 1

cos15 sin152 2

(c) 1 tan15

1 tan15

(ii)Find , to one decimal place, the elements of the set : 0 360x x

which satisfy the equation 2sin cos 60x x

5. Show that sin

tan tancos cos

A BA B

A B

and use it to find all the angles in the

range 0 180x which satisfy the equation tan tan3 tan 4x x x .

6. Determine all of the angles between 0 and 180 which satisfy the equation

cos5 cos3 sin3 sin

7. Express 3cos 4sinx x in the form cosR x , where R is positive and

lies between 0 and 90 .

What are the maximum and minimum values of 3cos 4sinx x , and for

what values of x in the range 0 to 360 do they occur?

Draw the graph of 3cos 4siny x x for values of x , at intervals of 30 , in

the range 0 to 360 , taking 1 cm to represent 30 on the x axis and 1 cm

to represent 1 unit on the y axis.

Using your graph, and giving solutions in range 0 to 360 :

(i) Solve the equation 6cos 8sin 7x x

(ii) Solve the inequality 6cos 8sin 7x x .

Common errors

1. Identities

Do not confuse identities and equations. An identity is true for all values of x

. An equation is only true for some values, and we find these values when we

solve the equation.

When proving an identity, do not assume the answer , i.e. do not start with the

expression you are asked to show true .Start with one side , and show that it

can be reduced to the other side. Alternatively , show that both sides van be

reduced to the same expression.

2. Solving equations

When you are solving an equation in the range 0 to360 , you must give all

the solutions. The trig functions take each value twice in that range.


When solving say sin 2 0.5x in the range 0 to 360 , you must consider

the values of 1sin 0.5 up to 720 .When you divide 2x by 2 the value will be

less than 360 .

When solving an equation of the form sin 0.5x in the range 0 to

360 , you may have to consider values of 1sin 0.5 greater than 360 , for

when is subtracted, the result may be within the correct range .

When you solve a quadratic in cos x , you might get the answer cos 2x or

1

2.Only the

1

2 value is relevant here-do not worry about the impossible value

of 2 .

3. Compound angles

These formulas must be used when finding sin or cos of compound angles.

Be aware of the following :

sin sin sinA B A B

Solution (to exercise)

10.1.2

2. 5 4 3

, ,3 5 4

3. 1 2

5, ,5 5

4. 2

2 2

1 2,

1 1

t t

t t

5. (a) 2sin x (b) 22sec x (c) cos sinx x (d) sin cosx x

7. (a) 2 2 9x y (b)2 2

14 9

x y (c) 2 2 1y x (d)

224 1

4

yx

(e) 2 2

1 2 1x y (f)

2 23 2

12 3

x y

(g) 2 2 1x y (h) 1xy

10.2.2

1. (a) sin 24 (b) cos 40 (c) tan 35 (d)

50cot

(e) sec10 (f) cos 64ec

2. (a) 35 ,325 (b)36 ,126 ,216 ,306 (c)103 ,317 (d)

86 ,314

(e) 76 ,284 (f) 92 ,188 (g) 44 (d) 215


(i) 24 ,216 (j)168 ,348

3. 143

4. 247

5. 15

17

6. 25

7

7. (a) 45 ,135 (b) 60 ,120 (c)55 ,125 (d)55 ,125

(e) 79 (f) 0 ,180 (g) 60 ,71 (h) 60

(i) 45

10.3.2

1. (a) sin 40 (b) cos 20 (c) sin A (d) cos 4A

2. (a)1

sin802

(b) tan 40 ]

3. (a) sin x (b) cos

4. (a)1 1 1

32 22

(b)

1 1 13

2 22

(c)

1 1 13

2 22

(d)

3 1

3 1

(e)1 1

12 2

(f) 2 1

5. 1

2

6. (a)56

65 (b)

16

65 (c)

63

65 (d)

56

33

7. (a) 0 ,180 ,32 (b) 39 (c) all angles (d) 45

(e) 0 ,180 ,30 ,150 (f) no solutions

10. (a) 65 ,175 (b)90 ,45 ,135 (c) 45 ,135 ,10 ,130 170

(d) 75 (e) 60 ,120 (f) 1 1

90 ,22 ,1122 2

11. 34cos 3cosA A

12. 33sin 4sinA A

10.4.2

1. (a) 5,53.1 (b) 41, 51 (c) 45,117 (d) 2, 45

(e)5

,2716

(f)145

, 561

2. (a) 5,37 (b) 41,141 (c) 45, 27 (d) 2,135

(e) 5

,6316

(f)145

,9561


3. (a) 5 5f (b) 41 41f (c) 45 45f

(d) 2 2f (e) 5.559 0.559f (f) 1.5052 1.5052f

4. (a) 40 ,108 (b) 79 , 157 (c) 9 ,73 (d)

96 , 170

5. 5, 53.1 . 5, 36.9 . 23 ,97r s x .

===========================================================

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




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