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.
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.
(b) Keying sequence Display
SHIFT sin ( 1 5 sin 35 ) EXE 6.587203533
)0.1nearest the to(cor.6.6
35sin5
1sin
Solution
(a) Keying sequence Display
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.
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
∴
d.p.) 1 to(cor.cm 4.24
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
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.
(Give your answers in fractions.)
(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
Follow-up question 6
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
By using a calculator, find the values of the following
expressions correct to 4 decimal places.
(a) cos 12.3 (b) 5
7cos 81
(c) 5
10 cos72 cos
(a) Keying sequence Display
cos 12.3 EXE 0.977045574
9770.03.12cos (cor. to 4 d.p.)
(b) Keying sequence Display
( 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
The answer is 60, i.e. = 60.
In degree mode, use the keys SHIFT and cos to find the corresponding acute angle .
Follow-up question 7
Solution
2
24 cos cos (b)
θ
0.474 cos (a) θ
Find the acute angle in each of the following using a
calculator. (Give your answers correct to 3 significant figures.)
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.
(b) Keying sequence Display
SHIFT cos ( 2 cos 75 ) EXE 58.8260478
)0.1nearest the to(cor.8.58
75cos2cos
Solution
(a) Keying sequence Display
SHIFT cos 0.583 EXE 54.33817552
degree)nearest the to(cor.54
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
Follow-up question 8
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
correct to 1 decimal place.
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
correct to 1 decimal place.
∵ EF
DEE cos
∴
d.p.) 1 to(cor.cm 8.3
cm 62cos8cm 8
62cos
DE
DE
Example 12
Solution
In △ PQR, ∠ P 36 , ∠ Q 90 and
PQ 10 cm. Find the length of PR
correct to 1 decimal place.
∵
PR
PQP cos
∴
d.p.) 1 to(cor.cm 4.12
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 )
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.
(Give your answers in fractions.)
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 .
Follow-up question 10
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
By using a calculator, find the values of the following
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:
The answer is 45, i.e. = 45.
In degree mode, use the keys SHIFT and tan to find the corresponding acute angle .
Find for a given value of tan
Follow-up question 11
2.77 tan (a) θ
Solution
Find the acute angle in each of the following using a
calculator. (Give your answers correct to 3 significant figures.)
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)
degree)nearest the to(cor.81
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
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
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.
Follow-up question 13
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
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.)
Follow-up question 14
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.
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
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
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°.
Follow-up question 15
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
Follow-up question 15 (cont’d)
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
Follow-up question 16
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)
Follow-up question 16 (cont’d)
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
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 .
By Pythagoras’ theorem,
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
Construct △ ABC as shown with 4
1 cos .
By Pythagoras’ theorem,
15
14 22
22
ABACBC
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.)
Construct △ ABC as shown with 9
40 tan .
By Pythagoras’ theorem,
41
1681
409 22
22
BCABAC
Solution
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?
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
Simplify the following expressions.
sintan(a
)(b)
(b)
2
22
cossincos
2tan1
2tan1
cos2 sin2 = 1
1
2
2
cossin
tan2 = (tan )2 and tan = cos _____ sin
Follow-up question 18
Solution
Simplify the following expressions.
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
(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
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 .
(Give your answers in fractions.)
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)
Using the trigonometric identities, find the acute angle in each of the following.
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
Using the trigonometric identities, find the acute angle in each of the following.
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
Follow-up question 19
Solution
(a)
(a)
(b)
42sin)90(cos 45tan35tantan
Using the trigonometric identities, find the acute angle in each of the following.
(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
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
∵
Follow-up question 20
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
Example 37Simplify the following expressions.
(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
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
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
correct to 1 decimal place.
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.)
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