Sine Ratio

Introduction to Trigonometric Ratios

θ

A

B C

AB is called the hypotenuse;

hypotenuse

BC is called the adjacent side of ;

AC is called the opposite side of .

adjacent side of

opposite side of

The figure below shows a right-angled triangle ABC, where B = and C = 90.

P

RQ

θθθθ

I only know that PQ2 + QR2 = PR2…

Consider the right-angled PQR

below. Is there any relationship among

, PQ, QR and PR?

How does the size of relate to the sides of the triangle?

In fact, the size of has certain relationship

between the ratios

These ratios are known as trigonometric ratios.

PQPR

PQQR,

QRPR

and .

Consider the following three right-angled triangles.

301

2

30

24

30

36

A B C

Complete the table below. What do you observe?

A B C

12

36

12=

24

12=hypotenuse

opposite side

Triangle

For a right-angled triangle with a given acute

angle ,

hypotenuse

opposite side is a constant.

Concept of Sine Ratio

hypotenuse

side opposite sin θ

θ

hypotenuse opposite side of

The sine ratio of an acute angle is defined as below:

301

2

302

4 3

30

6

sin 30 = 12

24

=36

=

For example,

For a right-angled triangle with a given acute angle , the sine ratio of is a constant.

5. and 13 ,90 , In ACABCABC

13

B

A

C

5

AC is the opposite side of B, and AB is the hypotenuse.

Follow-up question 1 . sin find figures, following the In θ

θ

15

8

17 θ

12

20

16

(a) (b)

Solution

1715

θ sin (a)

542016

θ sin (b)

Example 1

Solution

In △ PQR, ∠ P 90 , PQ 6, QR 10

and RP 8. Find the values of

(a) sin ∠ Q,

(b) sin ∠ R.

(Give your answers in fractions.)

(a)

5

410

8

sin

QR

PRQ

(b)

5

310

6

sin

QR

PQR

1. Make sure that the calculator is set in degree mode. Degree mode is usually denoted by the key DEG or D on calculators.

For example,

sin 30 EXE

The answer is 0.5.

Find sin for a given angle

Finding Sine Ratio Using Calculators

the value of sin 30 can be obtained by keying:

2. Use the key sin on a calculator to find the value of sin .

Follow-up question 2By using a calculator, find the values of the following expressions correct to 4 significant figures.

Solution

(a) sin 43 – sin 28

(b) 2 sin 11

(a) sin 43 – sin 28

(b) 2 sin 11

= 0.2125 (cor. to 4 d.p.)

= 0.3816 (cor. to 4 d.p.)

sin 43 = 0.681 99…, sin 28 = 0.469 47…

sin 11 = 0.190 80…

Example 2

Solution(a) Keying sequence Display

sin 66 EXE 0.913545457

9135.066sin (cor. to 4 d.p.)

By using a calculator, find the values of the following

expressions correct to 4 decimal places.

(a) sin 66 (b) sin 32.48

(b) Keying sequence Display

sin 32.48 EXE 0.537005176

5370.048.32sin (cor. to 4 d.p.)

Example 3

Solution

(b)

∵

)2634(sin26sin34sin

60sin26sin34sin

060sin26sin34sin

(a) By using a calculator, find the value of sin 34 + sin 26

sin 60 correct to 3 significant figures.

(b) From the result obtained in (a), is sin 34 + sin 26 equal

to sin (34 + 26 )?

(a) Keying sequence Display

sin 34 + sin 26 –sin 60 EXE 0.131538646

132.060sin26sin34sin (cor. to 3 sig. fig.)

In degree mode, use the keys SHIFT and sin to find the corresponding acute angle .

SHIFT sin 0.5 EXE

For example,

Find for a given value of sin

given that sin = 0.5, can be obtained by keying:

The answer is 30, i.e. = 30.

Follow-up question 3

Solution

Find the acute angle in each of the following using a

calculator. (Give your answers correct to 3 significant figures.)

(a) sin = 0.22

(b) sin = sin 68 – sin 40

(a) sin = 0.22

(b) sin = sin 68 – sin 40

= 12.7 (cor. to 3 sig. fig.)

= 16.5 (cor. to 3 sig. fig.)

= 0.2844…

sin 68 = 0.927 18…, sin 40 = 0.642 78…

Example 4Find the acute angles in the following using a calculator.

(a) sin 0.62, correct to the nearest degree.

(b) sin5

1 sin 35 , correct to the nearest 0.1 .

(c) 7 sin 3, correct to 3 significant figures.


SHIFT sin ( 1 5 sin 35 ) EXE 6.587203533

)0.1nearest the to(cor.6.6

35sin5

1sin

Solution


SHIFT sin 0.62 EXE 38.31613447

degree)nearest the to(cor.38

0.62sin

(c)

7

3sin

3sin7

Keying sequence Display

SHIFT sin ( 3 7 ) EXE 25.37693352

fig.) sig. 3 to(cor.25.4

7

3sin

Using Sine Ratio to Find Unknowns in Right-Angled TrianglesWe can use the sine ratio to solve problems involving right-angled triangles.

55

8 m

B

A

C

Find AC correct to 2 decimal places.

In ABC, C = 90, B = 55 and AB = 8 m.

Find Q correct to 2 decimal places.

In PQR, R = 90, PQ = 9 m and PR = 7 m.

Q

R

9 m

P

7 m

Follow-up question 4cm. 7 and90,75 , In ACCBABC

Find AB correct to 2 decimal places. 757 cm B

A

C

Solution

Example 5

Solution

In △ ABC, ∠B 90 , ∠C 42 and

AC 5 cm. Find the length of AB correct

to 1 decimal place.

∵ AC

ABC sin

∴

d.p.) 1 to(cor.cm 3.3

cm 42sin 5cm 5

42sin

AB

AB

Example 6

Solution

In △ ABC, ∠B 38 , ∠C 90 and

AC 15 cm. Find the length of AB correct

to 1 decimal place.

∵ AB

ACB sin

∴


cm 38sin

15

cm1538sin

AB

AB

Example 7In △ ABC, ∠ C 90 , AB 17 cm and

BC 9 cm. Find ∠ A correct to the

nearest degree.

∵

cm 17

cm 9

sin

AB

BCA

∴ degree)nearest the to(cor.32A

Solution

Cosine Ratio

Concept of Cosine Ratio

hypotenuse

side adjacent cos θ

θ

hypotenuse

The cosine ratio of an acute angle is defined as below:

adjacent side of

60

1

2

cos 60 = 12

60

2

4

60

3

6

24

=36

=

For example,

For a right-angled triangle with a given acute angle , the cosine ratio of is a constant.

5.2

B

A

C

2

In ABC, C = 90, AB = 5.2 and AC = 2.

AC is the adjacent side of A, and AB is the hypotenuse.

54108

θ cos (b)

Follow-up question 5 . cos find figures, following the In θ

θ5

4

3

θ8

10

6(a) (b)

Solution

53

θ cos (a)

Example 8

Solution

In △ PQR, ∠ P 90 , PQ 20, PR 21

and RQ 29. Find the values of

(a) cos ∠ Q,

(b) cos ∠ R.


(a)

29

20

cos

QR

PQQ

(b)

29

21

cos

QR

PRR

In degree mode, use the key cos to find the value of cos .

cos 30 EXE

For example,

The answer is 0.8660…

Find cos for a given angle

Finding Cosine Ratio Using Calculators

the value of cos 30 can be obtained by keying:

(b)2

75 cos40 cos


Solution

(a) 5 cos 29

By using a calculator, find the values of the following expressions correct to 3 significant figures.

29 cos 5 (a)2

75 cos40 cos (b)

= 4.37 (cor. to 3 sig. fig.)

cos 29 = 0.874 61…

= 0.512 (cor. to 3 sig. fig.)

cos 40 = 0.766 04…, cos 75 = 0.258 81…

Example 9

Solution


expressions correct to 4 decimal places.

(a) cos 12.3 (b) 5

7cos 81

(c) 5

10 cos72 cos


cos 12.3 EXE 0.977045574

9770.03.12cos (cor. to 4 d.p.)


( 7 5 ) cos 81 EXE 0.219008251

2190.081cos5

7 (cor. to 4 d.p.)

(c) Keying sequence Display

cos 72 – cos 10 5 EXE 0.112055443

1121.05

10 cos72cos

(cor. to 4 d.p.)

For example,

SHIFT cos 0.5 EXE

Find from a given value of cos

given that cos = 0.5, can be obtained by keying


In degree mode, use the keys SHIFT and cos to find the corresponding acute angle .


Solution

2

24 cos cos (b)

θ

0.474 cos (a) θ



0.474 cos (a) θ2

24 cos cos (b)

θ

cos 24 = 0.913 54…

Example 10Find the acute angles in the following using a calculator.

(a) cos 0.583, correct to the nearest degree.

(b) cos 2 cos 75 , correct to the nearest 0.1 . (c) 12 cos 5, correct to 3 significant figures.


SHIFT cos ( 2 cos 75 ) EXE 58.8260478


75cos2cos

Solution


SHIFT cos 0.583 EXE 54.33817552


583.0cos

(c)

12

5cos

5cos12

Keying sequence Display

SHIFT cos ( 5 12 ) EXE 65.37568165

fig.) sig. 3 to(cor.4.65

12

5cos

Using Cosine Ratio to Find Unknowns in Right-Angled TrianglesWe can use the cosine ratio to solve problems involving right-angled triangles.

Find BC correct to 2 decimal places.

In ABC, C = 90, B = 55 and AB = 8 m.

55

8 m

B

A

C

m 55 cos8 BC

Find Q correct to 2 decimal places.

In PQR, R = 90, PQ = 9 m and QR = 7 m.

9 m

Q

P

R7 m


cm. 3.5 and cm 4 ,90 , In BCABCABC

Find B correct to 2 decimal places.

Solution4 cm

B

A

C

3.5 cm

Example 11In △ DEF, ∠ D 90 , ∠ E = 62 and

EF = 8 cm. Find the length of DE

correct to 1 decimal place.

Example 12In △ PQR, ∠ P 36 , ∠ Q 90 and

PQ 10 cm. Find the length of PR


Example 13In △ PQR, ∠ R 90 , PQ = 22 cm

and QR =18 cm. Find ∠ Q correct to

the nearest 0.01 .

Example 11

Solution

In △ DEF, ∠ D 90 , ∠ E = 62 and

EF = 8 cm. Find the length of DE


∵ EF

DEE cos

∴


cm 62cos8cm 8

62cos

DE

DE

Example 12

Solution

In △ PQR, ∠ P 36 , ∠ Q 90 and

PQ 10 cm. Find the length of PR


∵

PR

PQP cos

∴


cm 36cos

10

cm 1036cos

PR

PR

Example 13

Solution

In △ PQR, ∠ R 90 , PQ = 22 cm

and QR =18 cm. Find ∠ Q correct to

the nearest 0.01 .

∵

cm 22

cm 18

cos

PQ

QRQ

∴ 35.10Q (cor. to the nearest 0.01 )

Tangent Ratio

Concept of Tangent Ratio

side adjacent

side opposite tan θ

θ

opposite side of

adjacent side of

The tangent ratio of an acute angle is defined as below:

45

1

1

tan 45 = 11

2

2

3

3

22

=33

=

45 45

For a right-angled triangle with a given acute angle , the tangent ratio of is a constant.

For example,

2.4

B

A

C3.2

In ABC, C = 90, AC = 2.4 and BC = 3.2.

AC is the adjacent side of A, and BC is the opposite side of A.

Follow-up question 9 . tan find figures, following the In θ

θ12

13

5(a) (b)

θ 3

5

4

Solution

512

θ tan (a)43

θ tan (b)

In △ PQR, ∠ R 90 , PQ 37,

PR 12 and RQ 35. Find the values

of

(a) tan∠ P,

(b) tan∠ Q.


Example 14

Solution

(a)

12

35

tan

PR

QRP

(b)

35

12

tan

QR

PRQ

For example,

The answer is 1.

tan 45 EXE

Find tan for a given angle

Finding Tangent Ratio Using Calculators

the value of tan 45 can be obtained by keying:

In degree mode, use the key tan to find the value of tan .


Solution

(cor. to 4 sig. fig.)

(cor. to 4 sig. fig.) 1.904

tan 51 = 1.234 89…(a) 7 tan 51

By using a calculator, find the values of the following expressions correct to 4 significant figures.

51 tan 7 (a)

57 tan

243 tan (b)

tan 43 = 0.932 51…, tan 57 = 1.539 86…

57 tan

243 tan (b)

Example 15

Solution


expressions correct to 4 significant figures.

(a) tan 28.26

(b) tan 65.32 tan 46.15

(a) 5375.026.28tan (cor. to 4 sig. fig.)

(b) 265.215.46tan32.65tan (cor. to 4 sig. fig.)

For example,

SHIFT tan 1 EXE

given that tan = 1, can be obtained by keying:


In degree mode, use the keys SHIFT and tan to find the corresponding acute angle .

Find for a given value of tan


2.77 tan (a) θ

Solution



tan 20 = 0.363 97…

20 tan3 tan (b) θ

Example 16

Solution

Find the acute angles in the following using a calculator.

(a) tan = 6.54, correct to the nearest degree.

(b) tan 2 tan 62 + 1, correct to 2 decimal places.

(c) 9 tan 2, correct to 4 significant figures.

(a)


54.6tan

(b)

d.p.) 2 to(cor.14.78

162tan2tan

(c) fig.) sig. 4 to(cor.53.12

2tan9

Using Tangent Ratio to Find Unknowns in Right-Angled TrianglesWe can use the tangent ratio to solve problems involving right-angled triangles.

Find AC correct to 2 decimal places.

In ABC, B = 50, C = 90 and BC = 12 m.

12 mB

A

C50

QR = 13 m. Find Q correct to 2 decimal places.

In PQR, R = 90, PR = 16 m and

16 m

Q

P

R

13 m


AC = 7 cm. Find B correct to 2 decimal places.

Solution

7 cmA

B

C

2.4 cm

Example 17In △ PQR, ∠ P = 65.2 , ∠ Q = 90 and

PQ = 4 cm. Find the length of QR

correct to 3 significant figures.

Example 18

In △ PQR, ∠ P = 90 , ∠ R = 42.6 and

PQ = 6.5 cm. Find the length of PR correct

to 3 significant figures.

Example 19In △ PQR, ∠ Q = 90 , PQ = 15 cm and

QR = 12 cm. Find ∠ P and ∠ R correct to

the nearest 0.1 .

Example 17

Solution

In △ PQR, ∠ P = 65.2 , ∠ Q = 90 and

PQ = 4 cm. Find the length of QR

correct to 3 significant figures.

∵ PQ

QRP tan

∴

fig.) sig. 3 to(cor.cm66.8

cm2.65tan4cm 4

2.65tan

QR

QR

Example 18

Solution

∵ PR

PQR tan

∴

fig.) sig. 3 to(cor.cm07.7

cm6.42tan

5.6

cm 5.66.42tan

PR

PR

In △ PQR, ∠ P = 90 , ∠ R = 42.6 and

PQ = 6.5 cm. Find the length of PR correct

to 3 significant figures.

Example 19In △ PQR, ∠ Q = 90 , PQ = 15 cm and

QR = 12 cm. Find ∠ P and ∠ R correct to

the nearest 0.1 .

Solution ∵

cm 15

cm 12

tan

PQ

QRP

∴ )0.1nearest the to(cor.7.38 P

∵

cm 12

cm 15

tan

QR

PQR

∴ )0.1nearest the to(cor.3.51 R

Simple Applications ofTrigonometric Ratios

Solving Problems Involving Plane Figures

A

B CD

35 40

8 cm

For plane figures involving right-angled triangles, we can use sine, cosine or tangent ratio to find the length of an unknown side or the size of an unknown angle.

Can you find the length of BC in the figure? Give your answer correct to 4 significant figures.

A

B CD

35 40

8 cmIn ABD, In ACD,

40 tanCDAD


Solution

605 cm

A

B CD

45.45 and 60 cm, 5

, of height the is figure, the In

ACBBADAB

ABCAD

Find AC.

(Give your answer correct to 3 significant figures.)

In ABD,

In ACD,

Example 20In the figure, BD is the height of △ ABC,

BC = 8 m, ∠ BAC = 70 and ∠ CBD = 60 . Find AC. (Give your answer correct to 2

decimal places.)

SolutionIn △ BCD,

m4

m60cos8

60cos

60cos

BCBDBC

BD

(1)m60sin8

60sin

60sin

BCDCBC

DC

In △ ABD,

(2)m70tan

4

70tan

70tan

BDAD

AD

BD

∴

d.p.) 2 to(cor.m38.8

m60sin870tan

4

DCADAC

from (1) and (2)

Draw PT QR as shown in the figure.

In △ PQT,

cm52sin6

52sin

52sin

PQPT

PQ

PT

Example 21

Solution

In the figure, PQRS is a trapezium with

∠ R = ∠ S = 90 , ∠ Q = 52 , PQ = 6 cm

and PS = 5 cm. Find the area of PQRS

correct to 2 decimal places.

cm52cos6

52cos

52cos

PQQT

PQ

QT

∴ Area of PQRS

d.p.) 2 to(cor. cm37.32

cm 52sin 65)52 cos 6(52

1

)(2

1

2

2

PTQRPS

The figure as shown is formed by a

right-angled triangle PQR and two

hemispheres with diameters PQ and RQ.

∠ PQR = 90 , ∠ PRQ = 25 and

PR = 16 cm. Find the perimeter of the

figure correct to the nearest cm.

Example 22

Solution

In △ PQR,

PR

PQ25sin

∴

cm 25sin16

25sin

PRPQ

PR

QR25cos

∴

cm 25cos16

25cos

PRQR

Perimeter of the figure

cm)nearest the to(cor. cm 49

cm 1625cos162

125sin 16

2

1

cm 162

1

2

1

QRPQ

Solving Real-life Problems

We can also use trigonometric ratios to solve real-life problems involving right-angled triangles.

Let’s study the example on the next page.

30 60B

A

C D

2 m

A rectangular advertising board is

fixed to a vertical wall and is

supported by two straight cable

wires AB and AC, as shown in the

figure. It is known that ABD = 30, ACD = 60 and CD = 2 m.

Find AC and AB.

(Give your answers correct to

3 significant figures if necessary.)

30 60B

A

C D2 m

In ACD,

30 60B

A

C D2 m

In ABD,


B

1 m

1.7 m

50D

A

C

A rectangular advertising board is fixed to

a vertical wall and is supported by two

straight cable wires AB and AC, as

shown in the figure. It is known that

m. 1.7 and m 1 ,50 BCACACB

(Give your answer correct to 3

significant figures.)

Find ABC.

Follow-up question 14 (cont’d)

Solution

1 m

1.7 m

50

A

B CD

In ADC,


Solution

1 m

1.7 m

50

A

B CD

In ABD,

(cor. to 3 sig. fig.)

Example 23There is a fish pond between A and B. A man

wants to go from A to B. He walks for 60 m

from A to C, then turns 75 clockwisely and

walks for 42 m from C to B. If ∠BAC = 30 , find AB. (Give your answer correct to 1

decimal place.)

Solution

Draw CD AB as shown in the figure.

In △ ACD,

AC

AD30cos

∴

(1)m 30sin60

30cos

ACAD

60

) of sum (3090180 △ACD

45

line) st.on s (adj.0675180BCD

In △ BCD,

BC

DBBCD sin

∴

(2)m 45sin42

sin

BCDBCDB

∴

d.p.) 1 to(cor.m7.81

m)45sin4230cos60(

DBADAB

Example 24

Find the values of the following

trigonometric ratios using the quarter of

the unit circle as shown. (Give your

answers correct to 1 decimal place.)

(a) sin 65 (b) sin 12

(c) cos 46 (d) cos 84

Solution

(a) Construct line segment OP such that

OP makes an angle 65 with the

positive x-axis.

9.0

of coordinate-65sin

Py

(b) Construct line segment OQ such that

OQ makes an angle 37 with the

positive x-axis.

2.0

of coordinate-12sin

Qy

(c) Construct line segment OR such that

OR makes an angle 46 with the

positive x-axis.

7.0

of coordinate-46cos

Rx

(d) Construct line segment OS such that

OS makes an angle 84 with the

positive x-axis.

1.0

of coordinate-84cos

Sx

Trigonometric Ratios of Special Angles

Can you find out the value of sin 60°?

In general, the values shown on the calculator screen are approximations only.

In fact, the exact values of the trigonometric ratios of some special angles such as 30°, 45° and 60° can be deduced from the properties of triangles.

With a calculator, I can evaluate sin 60° = 0.866 025 403...0.866025403

A

B C

First, let’s review on the trigonometric ratios and the Pythagoras’ theorem.

Consider right-angled triangle ABC, we have

c

a

bcb

B sin

ab

B tan

ca

B cos

1.

2. By Pythagoras’ theorem, 222 bac

Using the above knowledge and considering the following triangles,

A B

C

1

1

45°

45°

R

P Q2

60°

2 2

60°

60°

we can find the exact values of the trigonometric ratios of 30°, 45° and 60° .

Consider the isosceles right-angled triangle ABC on the right.Since B = 90°, we can apply Pythagoras’ theorem to find AC.

First, let’s find the exact values of the trigonometric ratios of 45° .

A B

C

1

1

45°

45°

(Pyth. theorem)______)()( 22 AC 1 1 2

22

or2

1

22

or2

1

1

2BC

ABBC

AB

ACBC

ACAB

We have sin 45° =

cos 45° =

tan 45° =

Trigonometric Ratios of 45°

Now, let’s try to find the trigonometric ratios of special angles 60° and 30° .

Then, find RS.

First, find PS and PRS.

First construct a perpendicular line from R and meet PQ at S.Consider the triangle PQR.We have found the exact values of the trigonometric ratios of 45° .Now, we can find the trigonometric ratios of 60° and 30°.

PQR is an equilateral triangle.

Since PRS and QRS are two congruent right-angled triangles,

PS = QS (corr. sides, s).

PSRP

S

____)()( 22 RS

____ and ___ PRSPS 1Consider △ PRS.

We have

(Pyth. theorem)

60 tan

60 cos

60 sin

30 tan

30 cos

30 sin

33

or3

1

23

21

23

21

3

12 3

30°

R

P Q

2 2

60°

60°

22

60°

PSRS

RPPS

RPRS

PRPS

PR

RS

RSPS

30°1

3

Trigonometric Ratios of 60° and 30°

θ

θ sin

θ cos

θ tan

30 45 60Trigonometric ratio

2

1

2

3

3

3 or

3

1

22

or2

1

2

2 or

2

1

1

2

3

2

1

3

They are useful when we need to find the values of trigonometric expressions involving special angles.

The table below summarizes the trigonometric ratios of the special angles 30°, 45° and 60°.

Without using a calculator, find the value of the expression sin 30° tan 60° + sin 60°.

23

23

3


Solution

Find the values of the following expressions without using a calculator.

60 tan1

30 tan (b) 22

60 cos45 cos45 sin

(a)

21

21

21

2121

60 cos45 cos45 sin

(a)

1

2

2

3

1

3

1

31

31

60 tan1

30 tan (b) 22

32


Solution

Find the values of the following expressions without using a calculator.

60 tan1

30 tan (b) 22

60 cos45 cos45 sin

(a)

For example:

1 2cos (a)

21

cos

60 cos 60° = __ 2 1

we can find the acute angles in simple trigonometric equations without using a calculator.

Since the exact values of the trigonometric ratios of special angles are known,

tan 45° = 1

45sin2tan2 (b)

22

2tan2

1tan

45

2 tan2


Solution

Find the acute angles in each of the following equations without using a calculator.

060sin tan21

(b) 45 cos sin2 (a)

45 cos sin2 (a)

22

sin2

21

sin

30

60

023

tan21

23

tan21

3tan

060sin tan21

(b)


Solution

Find the acute angles in each of the following equations without using a calculator.

060sin tan21

(b) 45 cos sin2 (a)

Example 25Find the values of the following expressions without using a calculator.

(a) cos 60 tan 30 tan 60

(b)

30 cos

30sin 445tan 2

(c) tan 60 sin 60 sin2 45

Solution

Example 26Find the acute angle in each of the following equations without

using a calculator.

(a) cos cos2 45

(b) 1)10( tan 3

Solution

Example 27Referring to the figure, find the lengths

of the following line segments without

using a calculator. (Leave your answers

in surd form.)

(a) AC (b) DC

Solution

Finding Trigonometric Ratios by Constructing Right-Angled Triangles

If sin = , how can I find cos

and tan ? Do I need to evaluate

first?

54

You can find cos and tan by the following steps without evaluating .

and AC = 5.

Step 3

Find the unknown side AB by Pythagoras’ theorem.

45

3

3

Step 245

Since sinθ= , we set

Step 1

Construct a right-angled triangle ABC with A =θand B = 90°.

4 5

3

22 BCACAB 22 45

opposite side of θ

BC = 4hypotenuse

Step 4

Find the other two trigonometric ratios by their definitions.

In general, if one of the trigonometric ratios of an acute angle θ is given, we can follow these steps to find the other two trigonometric ratios without evaluating θ.

ACABcos

ABBCtan

34

53

45

3

Follow-up question 17It is given that tanθ= 0.5, whereθis an acute angle. Find the values of sinθand cosθwithout evaluatingθ. (Give your answers in surd form.)

Solution

By Pythagoras’ theorem,

A B

C

1

2

22 ABACBC

22 21

5

Construct △ABC as shown with tanθ= .21

5.0tan 105

21

5

Follow-up question 17 (cont’d)It is given that tanθ= 0.5, whereθis an acute angle. Find the values of sinθand cosθwithout evaluatingθ. (Give your answers in surd form.)

Solution

By definition,

BCABcos

BCACsin

55

or 51

552

or 5

2

A B

C

1

2

5

Example 28

It is given that5

2 sin , where is an acute angle. Find the values of

cos and tan without evaluating . (Leave your answers in surd form.)

Construct △ ABC as shown with 5

2 sin .


21

25 22

22

BCACAB

Solution

Example 29It is given that cos 0.25, where is an acute angle. Find the values

of sin and tan without evaluating . (Leave your answers in surd

form.)

Solution

4

1100

25

25.0cos


1 cos .


15

14 22

22

ABACBC

By definition,

4

15

sin

AC

BC

15

1

15

tan

AB

BC

Example 30

It is given that9

40 tan , where is an acute angle. Find the value of

sin + cos without evaluating . (Give your answer in fraction.)


40 tan .


41

1681

409 22

22

BCABAC

Solution

Trigonometric Identities

I find thatI find that

30tan30cos30sin

45tan45cos45sin

60tan60cos60sin

160cos60sin

145cos45sin

130cos30sin

22

22

22

θ sin θ cos θ tan θ

30°

45°

60°

3

1

1

cos

sin 22 cossin

21

2

1

23

23

2

1

21

3

3

1

1

3

1

1

1

Basic Trigonometric Identities

Complete the table. What can you find?

Let’s study the following proof.

Is always true?

tancossin

Is always true? 1cossin 22

Consider the right-angled triangle ABC as shown.

ab

ca

cb tan,cos,sin

Then

cossin

cacb

tan

ab

(i)

Consider the right-angled triangle ABC as shown.

ab

ca

cb tan,cos,sin

22 cossin

Then

2

22

cab

(ii)22

ca

cb

c2 = b2 a2 (Pyth. theorem)2

2

cc

1

1cossin

cos

sin tan

22

θθ

θ

θθ

Note that

as writtenbe also can 1cossin 22 θθ

We have the following two basic trigonometric identities.

.sin1cos or

cos1sin 22

22

θθ

θθ

sintan(a

)

tan = sin cos _____

Simplify the following expressions.

cos1

(a)

(b)

sin1

tan

2tan1

sintan

12cos


sintan(a

)(b)

(b)

2

22

cossincos

2tan1

2tan1

cos2 sin2 = 1

1

2

2

cossin

tan2 = (tan )2 and tan = cos _____ sin


Solution


22 tan)sin1(

22 tancos

2

22

cossin

cos

22 tan)sin1( (a)

(a)

2

2

sin11cos

(b)

(b)

2

2

sin1)cos1(

2

2

cossin

2

2

sin11cos

2sin2tan

Example 31Simplify the following expressions.

(a)

cos tan

sin2

(b) 22 cos 4sin 4

(c) 4 cos 4

sin 362

2

Solution

(a)

cos

1 cos sin

sin

cos cos

sin sin

cos tan

sin

22

(b)

4

)1(4

)cos(sin4cos 4sin 4 2222

(c) 4cos 4

)cos1(36

4cos 4

sin 362

2

2

2

(∵ 1 cos sin 22 )

4

3

)1(cos4

)1(cos3

4cos 4

cos 33

2

2

2

2


(a)

2

2

tan

sin1

(b)

2sin1

cos sin

Solution

(a)

2

2

22

2

22

2

2

cos1

sin

cossin1

cos

sin

1sin1

tan

sin1

2sin (∵ 1 cos sin 22 )

(b)

tan cos

sincos

cos sin

sin1

cos sin22

Example 33

It is given that17

8 sin , where is an acute angle.

(a) By using the trigonometric identities, find the values of cos

and tan .

(b) Hence, find the value of

tan

cos 2 sin 5 .


Solution

15

817

1517

8 cos

sin tan

(b)

68

7515

817

1015

817

30

17

4015

817

152

17

85

tan

cos 2 sin 5

Trigonometric Ratios of Complementary AnglesConsider the right-angled triangle ABC as shown.

sin θ = sin (90° θ ) =

cos θ = cos (90° θ ) =

tan θ = tan (90° θ ) =

ba

ba

bc

bc

ca

ac

1

sin θ

cos θ

tan θ

sin (90° θ )

cos (90° θ )

tan (90° θ )

=

==

35 cos sin θ

35cossin

55

55sin

Using the trigonometric identities, find the acute angle in each of the following.

(a)

(b)

27tan

1)90( tan θ

cos = sin (90° )

(a)

(b)


27tan1

)90( tan θ

27tantan

27

tan (90° ) = tan _____ 1

35 cos sin θ(a)

(b)

27tan

1)90( tan θ

27tan1

tan1

)90( tan θ


Alternative Solution

27tan1

)90( tan θ

279090 θ

tan (90° ) = tan _____ 1

35 cos sin θ(a)

(b)

27tan

1)90( tan θ

(b)

)2790(tan

27


Solution

(a)

(a)

(b)

42sin)90(cos 45tan35tantan


(b)

42sin)90(cos

42sinsin

45tan35tantan

135tantan

35tan1

tan

)3590(tan

55tan

55

42

Example 34

It is given that7

24 tan , where is an acute angle. By using the

trigonometric identities, find the values of sin and cos . (Give your

answers in fractions.)

Solution∵

7

24 tan

∴

22 cos 576sin 49

(*) cos 24 sin 77

24

cos

sin

22 cos 576)cos1(49 (∵ 1 cos sin 22 )

25

7cos

625

49cos

49cos 625

2

2

∵ cos 24 sin 7 (from (*) )

∴

25

24 sin

25

7 24 sin 7

Example 35

It is given that3

1 cos , where is an acute angle. Using the

trigonometric identities, find the value of 3 cos2 sin2 .

(Give your answer in fraction.)

)cos1(cos 3sincos 3 2222 (∵ 1 cos sin 22 )

9

5

19

4

13

14

1cos 42

2

Solution

Proofs of Simple Trigonometric Identities

We have learnt five trigonometric identities.We can use them to prove other trigonometric identities.

(i)

(ii)

(iii)(iv)(v)

cos)90sin(

sin)90cos(

tan1

)90tan(

1cossin 22

cossin

tan

. tan cos

sin sin

1 that Prove

tan cos

.S.H.R

sinsin

1L.H.S.

sinsin1 2

tancos

tan1

cos

sincos2

sincos

cos

tancos

sinsin

1 1 sin2 = cos2

= cos sin _____

sin cos _____

1 ______

= tan _____ 1

∵


Solution

.tan)90(sin

11that Prove 2

2 θθ

tan.S.H.R 2)90(sin

11L.H.S.

2 θ

2cos1

1

2tan

2

2

cos)cos1(

2

2

cossin

R.H.S.L.H.S. ∵

22

tan)90(sin

11

Example 36

Find the acute angle in each of the following equations by using the

trigonometric identities.

(a)

66 tan

1 tan

(b) 2

sin)30( cos

Solution

(b)

290 cos)30( cos

2sin)30( cos

∴

40

602

32

9030


(a) )90( tan)90( cos)90( sin

(b) )90(sinsin )90( cos 2

Solution(a)

2cos

sin

cos sin cos

cos

sin1

sin cos

tan

1sin cos)90( tan)90( cos)90( sin

(b)

1

cossin

cos sin sin)90(sin sin)90( cos22

22

Example 38Find the values of the following expressions.

(a)

53sin

37sin12

2

(b) 21 sin 69 tan21 cos

Solution

(a)

153sin

53sin

53sin

)3790(sin

53sin

37cos

53sin

37sin1

2

2

2

2

2

2

2

2

(b)

0

21 cos21 cos

21 sin21 sin

21 cos21 cos

21 sin

21 cos

21 sin1

21 cos

21 sin21 tan

121 cos

21 sin)6990( tan

121 cos21 sin69 tan21 cos

Example 39Prove the following trigonometric identities.

(a) )90( sin sin

1

sin

cos

cos

sin

(b)

cos cos

1 tan sin

Solution(a)

cos sin

1 cos sin

cossin

sin

cos

cos

sinL.H.S.

22

cos sin

1

)90( sin sin

1R.H.S.

∵ L.H.S. R.H.S.

∴ )90( sin sin

1

sin

cos

cos

sin

(b)

tan sin.L.H.S

tan sin cos

sin sin

cos

sin

cos

cos1

cos cos

1R.H.S.

2

2

∵ L.H.S. R.H.S.

∴

cos cos

1 tan sin

Example 40

Prove that cos sin 21) cos (sin 2 .

cos sin 21

cos sin 2)cos(sin

cos cos sin 2sin

) cos (sinL.H.S.

22

22

2

R.H.S. cos sin 21

∵ L.H.S. R.H.S.

∴ cos sin 21) cos (sin 2

Solution

Extra Teaching Examples

Example 7 (Extra)

Solution

∵

cm 7.5

cm 5.5

sin

AB

ADB

∴ d.p.) 1 to(cor.2.47 B

In △ ABC, ∠ ADC = 90 , AB = 7.5 cm,

AC = 6.5 cm and AD = 5.5 cm. Find ∠ BAC,

∠ B and ∠ C correct to 1 decimal place.

∵

cm 6.5

cm 5.5

sin

AC

ADC

∴ d.p.) 1 to(cor.8.57 C

In △ ABC,

d.p.) 1 to(cor.0.75

8.572.47180

) of sum (180

△CBBAC

Example 13 (Extra)

Solution

In △ ABD, ∠D 90 , AB 11 cm, AC

7.8 cm and AD 6 cm. Find ∠BAC correct

to the nearest 0.1 .

∵

cm 8.7

cm 6

cos

AC

ADDAC

∴ 7.39DAC

∵

cm 11

cm 6

cos

AB

ADDAB

∴ 9.56DAB

∴


7.399.56

DACDABBAC

Example 19 (Extra)

Solution

In △ ABC, ∠ ABD = ∠ DBC = 25 , ∠ C = 90 and BC = 3.5 m. Find AD correct to 2 decimal

places.

In △ ABC,

∵ BC

ACABC tan

∴

m 50tan5.3m 5.3

)2525(tan

AC

AC

In △ BCD,

∵ BC

DCDBC tan

∴

m 25tan5.3m 5.3

25tan

DC

DC

∴

d.p.) 2 to(cor.m 54.2

m )25tan5.350tan5.3(

DCACAD

Example 21 (Extra)

Solution

The figure shows a quadrilateral ABCD

with ∠ A = 115 , ∠ B = 90 , ∠ C = 80 , AB =13 cm and AD = 18 cm. Find BC

correct to the nearest cm.

Draw DF BC and AE DF as shown in

the figure.

In △ ADE,

AD

AEDAE cos

∴

cm25cos18

cm)90115(cos18

cos

DAEADAE

∴ cm25cos18 AEBF ……(1)

AD

DEDAE sin

∴

cm25sin18

cm)90115(sin18

sin

DAEADDE

cm)1325sin18(

ABDE

EFDEDF

In △ CDF,

CF

DF80tan

∴

cm80 tan

1325 sin 1880 tan

DFCF

……(2)

∴

cm)nearest the to(cor. cm 20

cm80 tan

1325 sin 1825cos18

CFBFBC

Example 22 (Extra)

Solution

In the figure, sector OPQ is inscribed in

rectangle ABCD. Given that AB =10 cm and

BC = 14 cm, find the area of sector OPQ


In △ OBQ,

cm 14

cm 2

10

cos

OQ

OBQOB

∴ 1.69QOB

POAQOB

8.41

180)1.69(2

1802

line) st.on s (adj.180

POQ

POQ

QOBPOQ

POAQOBPOQ

∴ Area of sector OPQ

d.p.) 1 to(cor. cm 6.71

cm 14360

8.41

2

22

Example 27 (Extra)

The figure shows two shadows

AD and BD of a tree CD at

9:00 a.m. and 4:30 p.m.

respectively. If the height of the

tree is 8 m, find the distance

between A and B. (Leave your

answer in surd form.)

Solution

Distance between A and B

m )31(8

m )388(

DBAD

Example 38 (Extra)

Find the value of 26cos 364sin 26tan 33 222 .

0

)1(33

)26cos26(sin33

26cos 326sin 33

26cos 326cos26cos

26sin33

26cos 3)6490(cos26tan 33

26cos 364sin26tan 33

22

22

222

2

222

222

Solution

Example 39 (Extra)Prove that )90(tan sin)90(cos1 222 .

2

2

2

cos

sin1

) 90(cos1.L.H.S

2

2

22

2

22

22

22

cos

sin

cossin

cos

sin

1sin

tan

1sin

)90(tan sin.R.H.S

∵ L.H.S. R.H.S.

∴ )90(tan sin)90(cos1 222

Solution

Sine Ratio

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