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Advanced Mathematics Training Class Notes Chapter 8: Trigonometry

46

Chapter 8

Trigonometry

Trigonometry (三角學三角學三角學三角學)

Trigonometry is a branch of geometry that studies the relationship of triangles. Take a

simplest triangle – a right-angled triangle as example:

A

b

c

θ B a C

If we fix point B and C (the value of a), and vary the size of BCA∠ (value of θ), the value of b and c also changes. Actually, the value c/b is a function of θ. Trigonometry defined the

function “sine” (正弦) as followed:

sinc

bθ =

Similarly, there are also five other functions – cosine (餘弦), tangent (正切), cotangent (餘

切), secant (正割) and cosecant (餘割), together called the trigonometric functions (三角

函數):

cos , tan , cot , sec , csca c a b b

b a c a cθ θ θ θ θ= = = = =

For practical use, only sine, cosine and tangent are more useful.

A

Hypotenuse

Opposite

θ B Adjacent C

Respecting to θ, the side opposite to it is called the “opposite” (對邊), the one which is

longest is called “hypotenuse” (斜邊), and the one next to θ is called “adjacent” (鄰邊).

Therefore, the sine, cosine and tangent can also be defined as:

Opposite Adjacent Oppositesin , cos , tan

Hypotenuse Hypotenuse Adjacentθ θ θ= = =

It can be memorized by the mnemonic “oh, ah, oh-ah”, which represents, sin θ is opposite over hypotenuse, cos θ is adjacent over hypotenuse, tan θ is opposite over adjacent.

Definition of Angle

In plane geometry, the range of angle is 0° to 360°. However, this is not enough for practical

use. For example, a clock rotates more than 360°. It is necessary for us to redefine what is

angle.

In trigonometry, we interpret angle as rotation of rays (射線). When a ray OA rotates about O

to the position OB, the amount of rotation is defined as the angle AOB and is denoted by

AOB∠ . The angle is positive if it is rotated anti-clockwise, and is negative if is rotated

clockwise.

B

420° -75°

45°

O A

Base on the “new” definition of angle, we also have a more generalized definition of

trigonometric functions.


47

In a coordinate plane, take a point P (x, y). Let r denotes the distance from this point (P) to

the origin O (0, 0). (So 2 2r x y= + .) Let θ be the angle from the x-axis to OP.

P (x, y)

r

θ O

Then we define:

sin cos

tan cot

sec csc

y x

r r

y x

x y

r r

x y

θ θ

θ θ

θ θ

= =

= =

= =

These are the definition of the trigonometric functions of general angles θ.

Radians (弧度弧度弧度弧度)

In elementary mathematics, we measure angles in degrees (角度). Say, a right angle is 90°.

However, it is sometimes more convenient to use another measure: radians.

A

r s

θ O r B

In the figure, O is the center of a circle with radius r. A and B are two points on the

circumference. θ is the size of AOB∠ . s is the length of arc AB.

The radian measure of an angle is the ratio of the arc, which subtends the angle at the center

of a circle, and the radius of the circle. Or in formula,

s

rθ =

The following angle has a size of 1 radian.

r r

1 rad.

r

Usually the unit “rad.” is omitted, i.e., the size of the angle is 1.

Note that:

π radians = 180°

s

r θ

r

In a sector (扇形), if θ is measured in radian, then:

s = rθ

Area = 21 12 2r rsθ =


48

Basic Trigonometric Relations

Base on the definition of trigonometric functions, it is always true that:

1.

1csc

sin

1sec

cos

1cot

tan

θθ

θθ

θθ

=

=

=

(Reciprocal relations 倒數關係)

2.

sintan

cos

coscot

sin

θθ

θθ

θθ

=

= (Quotient relations 商數關係)

3.

2 2

2 2

2 2

sin cos 1

1 tan sec

1 cot csc

θ θ

θ θ

θ θ

+ =

+ =

+ =

(Squares relations 平方關係)

The last relations can be proved by Pythagoras’ theorem.

Some special angles

1 2 3 2

45° 60° 1 1

For 30°, 45° and 60°, their corresponding trigonometric functions will give some special

results. They are thus called the special angles. The results should be memorized, and is

summarized in the following table:

30° (ππππ/6) 45° (ππππ/4) 60° (ππππ/3)

sin 1

2

2 1or or 0.5

2 2

3or 0.75

2

cos 3

or 0.752

2 1

or or 0.52 2

1

2

tan 3 1

or3 3

1 3

Together with 0° and 90°, the value of sine and cosine shows an interesting pattern.

0° (0) 30° (ππππ/6) 45° (ππππ/4) 60° (ππππ/3) 90° (ππππ/2)

sin 0

2

1

2

2

2

3

2

4

2

cos 4

2

3

2

2

2

1

2

0

2

The followings should also be memorized:

0° (0) 90° (ππππ/2) 180° (ππππ) 270° (2ππππ/3) 360° (2ππππ) sin 0 1 0 -1 0

cos 1 0 -1 0 1

tan 0 / 0 / 0

“/” means undefined.

Extension: 1

sin181 5

° =+


49

Ranges of Trigonometric Functions

The output ranges of some trigonometric functions are constrained:

1 sin 1

1 cos 1

sec 1 or sec 1

csc 1 or csc 1

θθ

θ θ

θ θ

− ≤ ≤

− ≤ ≤

≤ − ≥

≤ − ≥

But tangent and cotangent are not. Their output range is any real numbers.

When 2

n πθ π= + , tangent and secant do not exist. When θ = nπ, cotangent and cosecant are

not defined.

In addition, their signs are also arranged in a special pattern:

0° < θθθθ < 90° 90° < θθθθ < 180° 180° < θθθθ < 270° 270° < θθθθ < 360°

0 < θθθθ < ππππ /2 ππππ/2 < θθθθ < ππππ ππππ < θθθθ < 3ππππ/2 3ππππ/2 < θθθθ < 2ππππ sin, csc + + – –

cos, sec + – – +

tan, cot + – + –

It can be simplified as:

90°

S A

180° 0° / 360°

T C

270°

“S” means “sine in positive”, “C” means “cosine is positive”, “T” means “tangent is

positive”, “A” means “all are positive”. This is known as the “CAST Diagram”.

Note: the part 0° to 90° is called quadrant I (第一象限), 90° to 180° is called quadrant II

(第二象限), and so on.

Transformation Formulae

Any angles can actually transformed into angles within range of 0° to 90° when dealing with

trigonometric functions.

Negative angles

( )( )( )

sin sin

cos cos

tan tan

θ θ

θ θ

θ θ

− = −

− =

− = −

Complementary angles

( )( )( )

2

2

12 tan

sin cos

cos sin

tan cot

π

π

πθ

θ θ

θ θ

θ θ

− =

− =

− = =

(Recall: π/2 = 90°)

Multiple revolutions

( ) ( )2f n fπ θ θ+ =

Where f can be sin, cos, tan, cot, sec or csc. n can be any integers.

Moreover, tan(πn + θ) = tan θ, cot(πn + θ) = cot θ.

Others

90° - θθθθ 90° + θθθθ 180° - θθθθ 180° + θθθθ 270° - θθθθ 270° + θθθθ 360° - θθθθ θθθθ

ππππ/2 - θθθθ ππππ/2 + θθθθ ππππ - θθθθ ππππ + θθθθ 3ππππ/2 - θθθθ 3ππππ/2 + θθθθ 2ππππ - θθθθ sin s c c s -s -c -c -s

cos c s -s -c -c -s s c

tan t k -k -t t k -k t

Here, “s” means “sin θ”, “c” means “cos θ”, “t” means “tan θ” and “k” means “cot θ”.


50

Compound Angle Formulae (複角公式複角公式複角公式複角公式)

We often encounter expressions like sin(A + B) or cos(A – B). If A is a multiple of π/2, then we can use the transformation formulae. But it’s not that easy. Fortunately, there are general

expressions of the trigonometric functions of sum or differences of angles.

( )( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

cot cot 1cot

cot cot

A B A B A B

A B A B A B

A BA B

A B

A BA B

A B

± = ±

± =

±± =

± =±

∓

∓

∓

These are called the “Compound Angle Formulae”. The first three formulae should be

memorized.

Example: Express tan 15° in surd form (根式) (with root signs)

Notice that tan 15° = tan (60° - 45°). Therefore,

( )tan15 tan 60 45

tan60 tan 45

1 tan60 tan 45

3 1

3 1

3 1 3 1

3 1 3 1

2 3

° = ° − °

° − °=

+ ° °

−=

+

− −=

+ −

= −

Double angle formulae (倍角公式倍角公式倍角公式倍角公式)

Replacing B with A, and use “+” for the “+” signs, we get the double angle formulae:

2 2

2

2

sin 2 2sin cos

cos2 cos sin

2 tantan 2

1 tan

cot 1cot 2

2cot

A A A

A A A

AA

A

AA

A

=

= −

=−

−=

For cosine, if we replace either cos2 A by 1 – sin2 A or sin2 A by 1 – cos2 A, we get two other

forms of double angle formulae for cosine:

cos 2A = 1 – 2 sin2 A = 2 cos2 A – 1

Triple angle formulae (三倍角公式三倍角公式三倍角公式三倍角公式)

Again, using compound angle formulae, we get the triple angle formulae: 3

3

sin3 3sin 4sin

cos3 4cos 3cos

A A A

A A A

= −

= −

The triple angle formulae can also be: 2 3

3 2

sin3 3cos sin sin

cos3 cos 3cos sin

A A A A

A A A A

= −

= −

But these are not as important as the two formulae given above.

Half angle formulae (半角公式半角公式半角公式半角公式)

Finally, the half angle formulae states that:

1 cossin

2 2

1 coscos

2 2

A A

A A

− = ±

+ = ±

The sign is determined by which quadrant does A/2 lie.


51

Product and Sum of Sine and Cosine

When solving trigonometric equations, we often come across with products or sums of sine

and cosine, which we may not be able to do easily. The product-to-sum formulae,

sum-to-product formulae and subsidiary angle form help us to tackle them.

Product-to-sum formulae (積化和差公式積化和差公式積化和差公式積化和差公式)

( ) ( )( )( ) ( )( )( ) ( )( )( ) ( )( )

12

12

12

12

sin cos sin sin

cos sin sin sin

cos cos cos cos

sin sin cos cos

A B A B A B

A B A B A B

A B A B A B

A B A B A B

= + + −

= + − −

= + + −

= − + − −

Sum-to-product formulae (和差化積公式和差化積公式和差化積公式和差化積公式)

( ) ( )( ) ( )( ) ( )( ) ( )

2 2

2 2

2 2

2 2

sin sin 2sin cos

sin sin 2cos sin

cos cos 2cos cos

cos cos 2sin sin

A B A B

A B A B

A B A B

A B A B

A B

A B

A B

A B

+ −

+ −

+ −

+ −

+ =

− =

+ =

− = −

The product-to-sum and sum-to-product formulae can be summarized as:

A B A+B A–B

2

2

2

2

s c s s

c s s s

c c c c

s s c c

= +

= −

= +

− = −

2

A B+ 2

A B− A B

Here, “s” means “sin”, “c” means “cos”.

Subsidiary angle form

Product-to-sum and sum-to-product mainly deal with sum or difference of two sines or

cosines. The subsidiary angle form is much difference. It changes expression in form of

sin cosa bθ θ+ to r cos(θ – α). Suppose ( )cos sin cosr a bθ α θ θ− = + . By expanding the left-hand-side, comparing

coefficients, we get:

sin

cos

a r

b r

αα

=

=

And therefore:

2 2

tan

r a b

a

bα

= +

=

Example: If ( )sin 3cos cosrθ θ θ α+ = − , find r and α.

( )221 3 2r = + =

( )6

1tan

3

30π

α

α

=

∴ = = °


52

Graphs of Trigonometric Functions (三角函數的圖像三角函數的圖像三角函數的圖像三角函數的圖像)

x

y

−π 0 π 2π

-1

1

y = sin x

x

y

−π 0 π 2π

-1

1

y = cos x

x

y

−π 0 π 2π

-2

2

y = tan x

x

y

−π 0 π 2π

-2

2

y = cot x

x

y

−π 0 π 2π

-2

2

y = sec x

x

y

−π 0 π 2π

-2

2

y = csc x


53

The previous page shows the graphs of the six trigonometric functions. In their graphs, we

see that they have repeating patterns. They are then called “periodic functions” (週期函數).

A function f(x) is periodic if and only if, there exists a real constant T > 0 such that:

f(x + T) = f(x) for all real values of x.

The smallest possible value of T is called the “period” (週期) of that function.

For sine, cosine, secant and cosecant, their periods are 2π. For tangent and cotangent, their periods are π.

Sine curve (正弦曲線正弦曲線正弦曲線正弦曲線)

The graph of y = sin(x) is exactly the same as the shape of waves. By this, we will go depth in

this type of curve.

The sine curve can be generalized as sinx d

y a cT

− = +

. In this expression, a is called the

“amplitude” (振幅) of the wave, T is the period, c presents the horizontal translation (橫

移), and d illustrates the vertical translation (縱移).

Note: The reciprocal (倒數) of T, 1/T, is sometimes called “frequency” (頻率)

The following graphs show how these four variables control the outlook of the curve.

x

y

−π 0 π 2π

-2

2

(Green curve: y = sin x. Red curve: y = 2 sin x. Pink curve: 12siny x= − )

x

y

0 π 2π 3π 4π

-1

1

(Blue curve: y = sin x. Green curve:2

sin xy = . Purple curve: y = sin 2x)

x

y

0 π 2π

-1

1

(Blue curve: y = sin x. Green curve: ( )4siny x π= − . Yellow curve: ( )5siny x π= + .)

x

y

0 π 2π

-2

2

(Blue curve: y = sin x. Green curve: y = sin x – 1. Purple curve: 32

siny x= + .)


54

Inverse Trigonometric Functions (反三角函數反三角函數反三角函數反三角函數)

Inverse function (反函數反函數反函數反函數)

For any function f(x), if y = f(x), there is a corresponding “inverse function” F(x) such that

x = F(y).

For example, if f(x) = 5x – 6, then its inverse function is ( ) 6

5

yF y

+= .

An inverse function of f(x) is usually denoted as f-1(x). In the above example,

( )1 6

5

xf x− +

= .

Inverse trigonometric function

Being functions, trigonometric functions also have their inverse functions. They are denoted

as sin-1 x, cos-1 x, etc. However, since trigonometric functions are all periodic, if y = f(x) (f is

a trigonometric function), there are multiple solutions of x. For example, if 12

sin x= , then

5 5 2 7 4 133 6 3 6 3 6 3 6, , , , , , , ,x π π π π π π π π= − − − … . But recall that a function can have one and only one

corresponding value of dependant variable. So we must take only one value out of so many

values. Which should we take? First, we define the principle value intervals (主值區間) of

these inverse functions. 1

2 2

1

1

2 2

sin

0 cos

tan

x

x

x

π π

π π

π

−

−

−

− ≤ ≤

≤ ≤

− < <

As π/6 is the only value inside range –π/2 to π/2, so 1 12 6

sin π− = .

Properties of inverse trigonometric function

By the principle of inverse function, there must be:

( ) ( )

( ) ( )

( ) ( )

1 1

1 1

1 1

sin sin sin sin

cos cos cos cos

tan tan tan tan

x x x x

x x x x

x x x x

− −

− −

− −

= =

= =

= =

Deriving from the relation #3 (Squares relations), we get:

( ) ( )( ) ( )( ) ( )

1 1 2

1 1 2

1 1 2

sin cos cos sin 1

sec tan csc cot 1

tan sec cot csc 1

x x x

x x x

x x x

− −

− −

− −

= = −

= = +

= = −

Or summarizing:

x sin cos tan cot sec csc

sin-1 x 21 x− 21

x

x−

21 x

x

− 2

1

1 x− 1

x

cos-1 21 x− x

21 x

x

− 21

x

x− 1

x

2

1

1 x−

tan-1 21

x

x+

2

1

1 x+ x

1

x 2

1 x+

21 x

x

+

cot-1 2

1

1 x+

21

x

x+ 1

x x

21 x

x

+ 2

1 x+

sec-1 2 1x

x

−

1

x 2

1x − 2

1

1x − x

2 1

x

x −

csc-1 1

x

2 1x

x

− 2

1

1x − 2

1x − 2 1

x

x − x

(The order of reading is first function, then inverse function, then “x”, and finally the cell)


55

General Solutions of Trigonometric Functions (三角函數通解三角函數通解三角函數通解三角函數通解)

We have learnt that if 12

sin x= , there are infinitely many solutions for x. Sometimes we are

told to find all of the solutions of the equation. Of course, we can’t list out that infinitely many solutions. We can use an expression with a free variable to represent it. The expression is called the general solution of a trigonometric function.

If s = sin x, then

( ) ( )

( ) ( )

1

1

1 sin In radian measure

180 1 sin In degree measure

n

n

x n s

x n s

π −

−

= + −

= ° + −

If c = cos x, then

( )( )

1

1

2 cos In radian measure

360 cos In degree measure

x n c

x n c

π −

−

= ±

= ° ±

If t = tan x, then

( )( )

1

1

tan In radian measure

180 tan In degree measure

x n t

x n t

π −

−

= +

= ° +

In the above expressions, n is any integer.

Example: 12

sin x= , then ( )6

1n

x n ππ= + − , where n is any integer.

In addition, if a range 0 < x < 2π (i.e., 0° < x < 360°) is specified, then

( )( )( )

1 1 1

1 1 1

1 1 1

sin or sin 180 sin

cos or 2 cos 360 cos

tan or tan 180 tan

x s s s

x c c c

x t t t

π

π

π

− − −

− − −

− − −

= − ° −

= − ° −

= + ° +

Slope and Trigonometry

Recall the definition of slope of a line:

1 2

1 2

y ym

x x

−=

−

C (x1, y1)

A (x2, 0) θ B (x1, 0)

x

If the second point lies on the x-axis (y2 = 0), the equation is reduced to 1

1 2

ym

x x=

−.

In the figure,

1 2

1

AB x x

BC y

= −

=

Therefore, in ∆ABC,

1

1 2

tanBC y

mAB x x

θ = = =−

That means,

tanm θ=

Where θ is the angle between the line and the x-axis, called the inclination (傾角)

Example: If slope of a line is 1, its inclination is 45°.


56

Angle between two lines

Consider two lines L1 and L2 having slopes m1 and m2 and inclination θ1 and θ2. We are

interested in the angle between them.

φ

L1 L2

θ1 θ2

( )( )

1 2

1 2

1 2

1 2

1 2

1 2

ext. of

tan tan

tan tan

1 tan tan

1

m m

m m

φ θ θ

φ θ θ

θ θθ θ

= − ∠

∴ = −

−=

+

−=

+

∵ △

Where φ is the angle between L1 and L2. Thus we got the formula:

1 2

1 2

tan1

m m

m mφ

−=

+

The absolute sign is there to ensure that an acute angle is selected.

Example: Find the angle between two lines with slope = -2 and 3 respectively.

( )( )2 3

tan1 2 3

1

45

φ

φ

− −=

+ −

=

∴ = °

Revision

In this chapter, we’ve learnt:

1. What are trigonometric functions

2. Definition of general angles

3. Radian measure of angles

4. Basic trigonometric relations

5. Trigonometric functions of some special angles

6. Ranges of trigonometric functions

7. Transformation formulae

8. Compound angle formulae

9. Product-to-sum and sum-to-product formulae, subsidiary angle form

10. Graphs of trigonometric functions

11. Inverse trigonometric functions

12. General solutions of trigonometric functions

13. Relationships between slope and tangent

14. Angle between two lines on coordinates plane


57

Exercise

In the followings, if not specified, x is the variable. Give answer in radians unless otherwise

stated. If general solution is required, use “n” for the free integral variable.

1. Express sin 75° the followings in surd form.

2. Find the general solutions of x in the following equations:

a) sin x + cos x = 1

b) 3 – cos2 x – 3 sin x = 0

c) ( ) ( )123 3

sin 4 cos xx π π −− =

3. Identify the amplitude, period, horizontal and vertical shift of the following sine curve:

x

y

−2π −π 0 π 2π

-2

2

4

4. Given θ = 54°. a) Prove that cos 2θ + sin 3θ = 0. b) Prove the sin 3β = 3 sin β – 4 sin3 β. c) Show that sin 54° is a root of the equation 4x3 + 2x2 – 3x – 1 = 0.

d) Find the exact value of sin 54°.

[Hint: Consider the sign of sin 54°]

5. (HKMO 2000 Heat) How many roots of θ are there in (cos2 θ – 1) (2 cos2 θ – 1) = 0, where 0° < θ < 360°?

6. (HKMO 1998 Heat) The circumference of a circle is 14π cm. Let X cm be the length of

an arc of the circle, which subtends an angle of 1/7 radian at the center. Find X.

7. (HKMO 2002 Heat) Given that A∠ is a right angle in ∆ABC, 2 2 14

sin cosC C− = and

40AB = . Find the length of BC.

8. (PCMSIMC 2001) If sin x° = cos x°, where 0 < x < 360, find the sum of all possible

values of x.

9. (PCMSIMC 2001) Evaluate 89

1

tan

1 tank

k

k=

°+ °∑

10. (HKCEE 1995 Partial) Answer the following questions.

a) Show that cos2 A – cos2 B = sin(A + B) sin(A – B)

b) Show that cos2 x – sin2 y = cos(x + y) cos(x – y)

c) Find the general solutions of 2 2cos 2 cos sin5 sin 3 0x x x x+ − =

11. Prove that (sin x + cos x)2 = 1 + sin 2x

[Note: This identity worth memorizing]

12. Prove that 1

0

2cos 0

n

k

kx

n

π−

=

+ =

∑ .

13. Prove that 1

0

2sin 0

n

k

kx

n

π−

=

+ =

∑

14. If tan a and tan b are roots of x2 + 2x + 3 = 0, find tan (a + b).

15. The vertices of ∆ABC are A (2, -1), B (7, 1) and C (-1, 5). Find tan A + tan B + tan C. 16. (IMO Prelim HK 2003) Find (cos 42° + cos 102° + cos 114° + cos 174°)2.


58

Suggested Solutions for the Exercise

1) 2 6

4

+

2a) 2

2 or 2n nππ π+

b) ( )2

1n

n ππ + −

c) 1 7

4 12n

ππ +

3) a = 3, T = 1/2, c = π/2, d = 1.

4d) 5 1

4

+

5) 5

6) 1

7) 8

8) 270

9) 44.5

10c) 2 5 20

2 or nn π π ππ ± −

14) 1

15) 111/8

16) 3/4

trigonometry - yuen long merchants association secondary...

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