7/18/2020 chapter 7
TRANSCRIPT
7/18/2020 Chapter 7. INTRODUCTION TO
TRIGNOMETRY
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MATHEMATICS Science Group 10th
Contents Exercise 7.1 .............................................................. 1
Exercise 7.2 .............................................................. 4
Exercise 7.3 .............................................................. 9
Exercise 7.4 ............................................................ 13
Exercise 7.5 ............................................................ 16
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Radian:
The angle subtended at the Centre of the circle by an
arc, whose length is equal to the radius of the circle is
called one radian.
Relationship between radians and degree:
Things to know:
So,
1 πππ£πππ’π‘πππ = 2π ππππππ = 3600
2π ππππππ = 3600 π‘π πππ‘ 1 ππππππ πππ£πππ ππ¦ 2π
1 ππππππ =3600
2π
1 ππππππ =1800
π
To get 1 degree divide by 360 2π
360πππ = 10
10 =π
180πππ
Remember that:
1 radian=180
3.1416= 57.295795π β 57π17β²45β²β²
1π =3.14516
180= 0.0175 πππππππ
Exercise 7.1 Q.1: Locate the following angles:
i. 300
ii. 221
2
0
iii. 1350
iv. β2250
v. β600
vi. β1200
vii. β1500
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viii. β2250
Q.2:
Express the following sexagesiml measures of angles
in decimal form.
i. 450 30β² Solution:
= 45π +300
600
= 45π + 0.50
= 45.5π
ii. 60030β²30β²β² Solution:
= 600 +30π
600+
30π
60π Γ 60π
= 60π + 0.5π + 0.0080
= 60.5080
iii. 125π22β²50β²β²
Solution:
= 125π +22π
60π+
50π
60π Γ 60π
= 1250 + 0.3670 + 0.0139π
= 125.38080
Q.3: Express the following in π·ππβ²πβ²β²:
i. 47.360 Solution:
= 470 + 0.360
= 470 + (0.36 Γ 60)β²
= 470 + 21β² + (0.6 Γ 60)β²β²
= 470 + 21β² + 36β²β²
= 47021β²36β²β²
ii. 125.45π Solution:
= 1250 + 0.450
= 1250 + (0.45 Γ 60)β²
= 2250 + 27β²
225027β²0β²β²
iii. 225.75π Solution:
= 225π + π. 75π
= 125π + (0.75 Γ 60)β²
= 1250 + 45β² = 225π45β²0β²β²
iv. β22.50
Solution:
= β[220 + 0.50]
= β[220 + (0.5 Γ 60)β²] = β[22π + 30β²]
= β22030β²
v. β67.580 Solution:
β(67π + 0.580)
= β[670 + (0.58 Γ 60)β²]
= β[670 + 34β² + 0.8β²]
= [670 + 34β² + (0.8 Γ 60)β²β²]
= β[670 + 34β² + 48β²β²] = β67π34β²48β²β²
vi. 315.180 = 315π + 0.18π
= 3150 + (0.18 Γ 60)β² = 315 + 10.8β²
= 315π + 10β² + (0.8 Γ 60)β²β²
= 3150 + 10β² + 48β²β² = 315π10β²48β²β²
Q.4: Express the following angles into radians.
i. 300
= 30π
180πππππππ
= 30π
30 Γ 6πππππππ
=π
6 πππππππ
ii. 600
= 60 Γπ
180 ππππππ
= 60π
60 Γ 3ππππππ
=π
3πππππππ
iii. 135π Solution:
2250
= 225 π
180πππππππ
= 45 Γ 3π
45 Γ 4πππππππ
=3π
4πππππππ
iv. 2250
Solution:225π
= 225 π
180πππππππ
= 45 Γ 5π
45 Γ 4πππππππ
=5π
4πππππππ
v. β1500 Solution:
β1500
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= β150 π
180πππππππ
= β5 Γ 30π
30 Γ 6πππππππ
=β5π
6πππππππ
vi. β2250
Solution:
= β225 π
180πππππππ
= β5 Γ 45π
45 Γ 4πππππππ
=β5π
4πππππππ
vii. 3000 Solution:
= 300 π
180πππππππ
= 60 Γ 5π
60 Γ 3πππππππ
=5π
3πππππππ
viii. 3150 Solution:
= 315 π
180πππππππ
= 45 Γ 7π
45 Γ 4πππππππ
=7π
4πππππππ
Q.5: Convert each of the following to degrees.
i. 3π
4
Solution: 3π
4πππππππ
=3π
4 180
π ππππππ
=3π
4
180
π ππππππ
= 3 Γ 45 πππππππ
= 135π
ii. 5π
6
Solution: 5π
6πππππππ
=5π
6 180
π ππππππ
=5π
6
180
π ππππππ
= 5 Γ 30 πππππππ
= 150π
iii. 7π
8
Solution: 7π
8πππππππ
=7π
8 180
π ππππππ
= 7 Γ 180
8 ππππππ
=1260
8 πππππππ
= 157.5π
iv. 13π
16
Solution: 13π
16πππππππ
=13π
16 180
π ππππππ
= 13 Γ 180
16 ππππππ
=2340
16 πππππππ
= 146.25π
v. 3 πππππππ Solution:
3 πππππππ
= 3 180
π ππππππ
=540
π πππππππ
= 171.887π
vi. 4.5
Solution:
4.5 πππππππ
= 4.5 180
π ππππππ
=810
π πππππππ
= 257.8310
vii. β7π
8
Solution:
β7π
8 πππππππ
= β7π
8 180
π ππππππ
=β1260
8 πππππππ
= 157.5π
viii. β13
16π
Solution:
β13π
16 πππππππ
= β13π
16 180
π ππππππ
=β2340
16 πππππππ
= 146.25π
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Sector of a Circle:
i. Arc of a circle:
A part of the circumference of a circle is called
an arc.
ii. Segment of a circle:
A part of the circular region bounded by an arc
and a chord is called segment of a circle:
iii. Sector of a circle:
A part of a circular region bounded by the two
radii and an arc is called sector of the circle.
Area of the Circular sector:
Consider a circle of radius and an arc of length
units, subtended an angle π ππ‘ π.
Exercise 7.2 Question No.1 Find π½ ππππ βΆ
i. π° = πππ, π = π. πππ Solution: using rule
πΌ = ππ
2 = 3.5π 2
3.5= π
π = 0.57 ππππππ
ii. π° = π. ππ , π = π. π π Solution: πππππ ππππ
πΌ = ππ
πΌ
π= π
4.5
2.5= π
π = 1.8ππππππ
Question No.2 find π° ππππ
i. π½ = ππππ, π = π. πππ
Solution:
π΄π π π βππ’ππ ππ ππ πππππ’π π π
π = 1800
= 180 π
180 ππππππ
= π ππππππ
Using rule πΌ = ππ
= 4.9ππ Γ π
= 15.4ππ
ii. π½ = πππ ππβ², π = ππππ
Solution:
π΄π π π βππ’ππ ππ ππ ππππππ, π π
π = 60π 30β²
= 600 +300
60π
= 60.5π
= 60.5 π
180ππππππ
π = 1.056 ππππππ
π = 1.056 ππππππ
π’π πππ ππ’ππ πΌ = ππ
= 15ππ Γ 1.056
= 15.84ππ
Questions No.3 find π, π€βππ
i. πΌ = 4ππ, π =1
4 ππππππ
Solution:
ππ πππ π π’ππ πΌ = π π
4ππ = π1
4
4ππ Γ 4 = π
π = 16ππ
ii. πΌ = 52ππ , π = 45π
Solution:π΄π π π βππ’ππ ππ ππ πππππππ .
π = 45π
= 45π
180 ππππππ
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=π
4ππππππ
Now using rule πΌ = ππ
52ππ = ππ
4
52ππ Γ 4
π= π
π = 66.21 ππ
Question No.4 In a circle of radius 12cm, find the
length of an arc which subtends a
Central angle π½ = π. π πππ πππ
Solution:
π ππππ’π = π = 12ππ
π΄ππ πππππ‘β =?
πΆπππ‘πππ πππππ = π = 1.5 ππππππ
Using rule πΌ = ππ
πΌ = 12π Γ 1.5
πΌ = 18π
Question No.5 In a circle of radius 10m, find the
distance travelled by a point moving on this circle if
the point makes 3.5 revolution.
Solution:
π ππππ’π = π = 10π
ππ’ππππ ππ πππ£πππ’π‘ππππ = 3.5
π΄ππππ ππ πππ πππ£πππ’π‘πππ = 2π
π΄ππππ ππ 3.5 πππ£πππ’π‘πππ = π
= 3.5 Γ 2 πππππππ
π = 7π ππππππ
Distance travelled= πΌ =?
ππ πππ ππ’ππ πΌ = ππ
πΌ = 10π Γ 7π
πΌ = 220π
Question No.6 What is the circular measure of the
angle between the hands of the watch at 3 Oβ clock?
Solution:
At 3 πβ² clock the minute hand will be at 12 and hour
hand will be at 3 i.e the angle between the hands of
watch will be one quarter of the central angle of full
circle.
π. π =1
4 ππ 360π
1
4Γ 360π
= 90π
= 90π
180ππππππ
=π
2ππππππ
Question No.7 What is the length of arc APB?
Solution:
From the figure we see that
π ππππ’π = π = 8ππ
πΆπππ‘πππ πππππ = π
= 900
=π
2ππππππ
Arc length πΌ =?
By rule πΌ = ππ
πΌ = 8ππ Γπ
2
πΌ = 4ππ Γ π
πΌ = 12.57 ππ
So, length of arc APB is 12.57 cm
Question No.8 In a circle 12cm, how long an arc
subtended a central angle of πππ?
Solution:
Radius = π = 12ππ
π΄ππ πππππ‘β = πΌ =?
πΆπππ‘πππ πππππ = π = 840
= 84π
180 ππππππ
= 1.466 ππππππ
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Now by rule πΌ = ππ
12ππ Γ 1.466
= ππ. π ππ
Question No.9 Find the area of sector OPR
(a)
Radius= π = 6ππ
Central angle = π = 60π
= 60π
180 ππππππ
=π
3ππππππ
Area of sector =?
As area of sector =1
2π2π
=1
2Γ (6ππ)2 Γ
π
3
=1
6Γ 36ππ2 Γ π
= 6π ππ2
= 18.85ππ2
(b)
Radius = π = 20ππ
πΆπππ‘πππ πππππ = π = 450
= 45π
180 ππππππ
=π
4ππππππ
Area of sector =?
Area of Sector =1
2π2π
=1
2(20ππ)2 Γ
π
4
=400ππ2
8Γ π
= 50πππ2
= 157.1 ππ2
Question No.10 Find area of sector inside a central
angle of πππ ππ π ππππππ ππ πππ πππ ππ.
Solution:
Area of sector =?
π ππππ’π = π = 7π
Central angle= π = 20π
= 20π
180 ππππππ
=π
9ππππππ
π΄πππ ππ π πππ‘ππ =1
2π2π
=1
2Γ (7π)2 Γ
π
9
=49π
18π2
= 8.55π2
Question No.11 Sehar is making skirt. Each panel of
this skirt is of the shape shown shaded in the
diagram. How much material (cloth) is required for
each panel?
Solution:
Central angle = π = 800
= 80 π
180ππππππ
=4π
9 ππππππ
Radius of bigger sector = π = (65 + 10)ππ
π = 66ππ
Radius of smaller sector= π = 10ππ
πβππππ ππππ =?
π΄πππ ππ ππππππ π πππ‘ππ =1
2π 2π
=1
2Γ (66ππ)2 Γ
4π
9
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= 4356ππ2 Γ2π
9
968πππ2
Area of smaller π πππ‘ππ =1
2 π2π
=1
2π2π
=1
2(10ππ)2 Γ
4π
9
=200
9πππ2
Shaded area 968π β200
9π
=8712π β 200π
9
=8512
9πππ2
= 2971.25ππ2 Question No.12 Find the area of a sector with central
angle of π
π radian in a circle of radius 10cm.
Solution:
Area of sector =?
πΆπππ‘πππ πππππ = π =π
5 ππππππ
π ππππ’π = π = 10ππ
Area of sector =1
2π2π
=1
2(10ππ)2 Γ
π
5
=1
10Γ 100ππ2 Γ π
=1
10Γ 100ππ2 Γ π
= 10πππ2
= 31.43 ππ2 Question No.13 The area of sector with central angle
π½ in circle of radius 2m is 10 square meter. Find
π½ ππ πππ πππ
Solution:
Area of sector = 10π2
Radius= π = 2π
Central angle= π =?
As ππππ ππ π πππ‘ππ =1
2π2π
10π2 =1
2(2π)2π
10π2 =1
2(4π2)π
10π2 = 20π2
π =10π2
2π2
π = 5 ππππππ
Angle in Standard position:
A general angle is said to be in standard position if its
vertex is at the origin and its initial side is directed
along the positive direction of the π₯ β ππ₯ππ of a
rectangular coordinates system.
The position of the terminal side of a angle in
standard position remains the same if measure of an
angle is increased or decreased by a multiple of 2π
Some Standard angles are shown in the following
figures.
The Quadrants and Quadrantal Angles:
The π₯ β ππ₯ππ and π¦ β ππ₯ππ divides the plane in four
regions. Called quadrants, When they intersect each
other at right angle. The point of intersection is called
origin and is denoted by O.
Angle between ππand 90π are in the first
quadrant.
Angle between 90πand 180π are in the
second quadrant.
Angle between 180πand 270π are in the
third quadrant.
Angle between 270πand 360π are in the
fourth quadrant.
An angle in standard position is said to lies in
that quadrant if its terminal side lies in that
quadrant. Angles πΌ, π½, πΎ πππ π lie in
πΌ, πΌπΌ , πΌπΌπΌ πππ πΌπ quadrant respectively.
Quadrantal Angles:
If the terminal side of an angle in standard position
falls on π₯ β ππ₯ππ or π¦ β ππ₯ππ then it is called a
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quadrantal angles. The quadrantal angles are shown
as below.β
Trigonometric Ratios and their reciprocals with the
help of a unit circle:
There are six fundamental trigonometric ratios called
sine, cosine, tangent, cotangent ,secant and cosecant.
To define these functions we uses circular approaches
which involves the unit circle.
Let π be real number, which represents the radian
measure of an angle in standard position. Let π(π₯, π¦)
Be any point on the unit circle lying on terminal side
of π as shown in figure.
We define sine of π, π€πππ‘π‘ππ ππ π πππ πππ πππ πππ ππ
π written as
π πππ =πΈπ
ππ=
π¦
1= π πππ = π¦
πππ π =ππΈ
ππ=
π₯
1 πππ π = π₯
π. π cos π πππ π πππ πππ π‘βπ π₯ βcoordinates and
π¦ βcoordinates of the point P on the unit circle. The
equations π₯ = πππ π and π¦ = π πππ are called circular
or
Trigonometric functions.
The remaining trigonometric functions tangent,
cotangent, sector and cosecant will be denoted by
π‘πππ , πππ‘π, π πππ, πππ πππ πππ πππ πππ¦ ππππ πππππ π.
ππππ =πΈπ
ππΈ=
π¦
π₯ π‘πππ =
π¦
π₯(π₯ β 0)
π΄π π¦ = π πππ πππ π₯ = πππ π π‘πππ =π πππ
πππ π
πππ‘π =π₯
π¦(π¦ β 0) πππ‘π =
πππ π
π πππ
π πππ =1
π₯(π₯ β 0) πππ πππ πππ =
1
π¦(π¦ β 0)
π πππ =1
πππ π πππ πππ πππ =
1
π πππ
Reciprocal Identities:
Sin π =1
πππ πππ ππ πππ πππ =
1
sin π
πππ π =1
π πππ ππ π πππ =
1
πππ π
π‘πππ =1
πππ‘π ππ πππ‘π =
1
π‘πππ
Signs of trigonometric ratios in different Quadrants:
Allied angles:
sin(βπ) = βπ πππ cos(βπ) = πππ π
sin(90 β π) = πππ π cos(90 β π) = π πππ
sin(90 + π) = πππ π sin(90 + π) = βπ πππ
sin(180 β π) = π πππ cos(180 β π) = βπππ π
sin(180 + π) = βπ πππ cos(180 + π) = βπππ π
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Exercise 7.3 Question No.1 Locate each of the following angles in
standard position using a protector or fair free hand
guess, also find a positive and a negative angle
conterminal with each given angle:
Solution:
i. 170π Positive coterminal angle = 360π + 170π
= 530π
Negative coterminal angle = β1900
ii. 7800
Positive coterminal angle 7800 +2[360π] =
600
Negative coterminal angle = β3000
iii. β1000
Positive coterminal angle 2600
Negative coterminal angle = β360π β 100π
= β4600
iv. β500π
Positive coterminal angle = 2200
Negative coterminal angle= β140π
Question No.2 Identity closest quadrantile angles
between which the following angles lie.
i. 1560
Answer: 900 πππ 1800
ii. 3180
Answer: 2700 πππ 3600
iii. 5720
Answer: 5400 πππ 6300
iv. β3300 Answer: 0π πππ 90π
Question No.3 Write the closest quadrantal angles
between which the angles lie. Write your answer in
radian measure.
i. π
3
Answer : π ππππ
2
ii. 3π
4
Answer: π
2 πππ π
iii. βπ
2
Answer: 0 πππ βπ
2
iv. β3π
4
Answer : βπ
2 πππ β π
Question No.4 in which quadrant π½ lies, when
i. π πππ > 0, π‘ππ < 0 Answer πΌπΌ ππ’ππππππ‘
ii. πππ π < 0, π πππ < 0 Answer: πΌπΌπΌ ππ’ππππππ‘
iii. π πππ > 0, π πππ < 0
Answer : πΌπ ππ’ππππππ‘
iv. πππ < 0 , π‘πππ < 0 Answer:πΌπΌ ππ’ππππππ‘
v. πππ πππ > 0, πππ π > 0 Answer:πΌ ππ’ππππππ‘
vi. π πππ < 0, π πππ < 0 Answer: πΌπΌπΌ ππ’ππππππ‘
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Question No.5 Fill in the blanks:
i. cos(β1500) = _________ πππ 150π
ii. sin(β3100) = _________ π ππ310π
iii. tan(β2100) = _________ π‘ππ210π
iv. cot(β450) = _________ πππ‘450
v. sec(β600) = _________ π ππ60π
vi. cosec(β1370) = _________ πππ ππ137π
Answers:
i. +π£π ii. βπ£π iii. βπ£π iv. βπ£π v. +π£π vi. βπ£π
Question No.6 The given point p lies on the
terminal side of π½, Find quadrant of π½ and all six
trigonometric ratios.
i. (β2,3) π€π βππ£π π₯ = β2 and π¦ = 3, π π π lies in
quadrant πΌπΌ.
π = βπ₯2 + π¦2
= β(β2)2 + (3)2
= β4 + 9
= β13
Thus,
π πππ =π¦
π=
3
β13 πππ πππ =
β13
3
πππ π =π₯
π= β
2
β13 π πππ = β
β13
2
π‘πππ =π¦
π₯= β
3
2 πππ‘π = β
2
3
ii. (β3,4)
π€π βππ£π π₯ = β3 and π¦ = 4, π π π lies in
quadrant πΌπΌπΌ.
π = βπ₯2 + π¦2
= β(β3)2 + (4)2
= β9 + 16
= β25 = 5
Thus,
π πππ =π¦
π=
β4
5 πππ πππ =
β5
4
πππ π =π₯
π=
β3
5 π πππ = β
5
3
π‘πππ =π¦
π₯=
4
3 πππ‘π =
3
4
iii. (β2, 1)
We have π₯ = β2 πnd π¦ = 1 so π lies in quadrant πΌπΌ.
π = βπ₯2 + π¦2
π = β(β2)2
+ (1)2
= β2 + 1
= β3
Thus,
π πππ =π¦
π=
1
β3 πππ πππ = β3
πππ π =π₯
π=
β2
β3 π πππ =
β3
β2
π‘πππ =π¦
π₯=
1
β2 πππ‘π = β2
Question No.7 if ππ¨π¬ π½ = βπ
π and terminal arm of
the angle π½ is in quadrant π°π°, find the valves of
remaining trigonometric functions.
In any right triangles πππ
cosπ = β2
3=
π₯
π π‘βππ π₯ = β2 πππ π = 3
Also,
π πππ =1
πππ π= β
3
2
As we know
π2 = π₯2 + π¦2 (3)2 = (β2)2 + π¦2
9 = 4 + π¦2
π¦2 = 5
π¦ = Β±β5 π π π¦ = β5
Now
π πππ =π¦
π=
β5
3 πππ πππ =
π
π¦=
3
β5
πππ π =π₯
π=
β2
3 π πππ =
π
π₯=
β3
2
π‘πππ =π¦
π₯=
ββ5
2 πππ‘π =
β2
β5
Question No.8 ππ ππππ½ =π
π πππ ππππ½ <
π, ππππ πππ valves of other trigonometric functions
at π½
Solution:
As tanπ =3
4 πππ π πππ ππ β π£π, π€βππβ is possible in
quadrant πΌπΌπΌ only. We complete the figure.
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From the figure π₯ = β3 πππ π¦ = β4
By Pythagorean theorem
π2 = π₯2 + π¦2
π = βπ₯2 + π¦2
π = β(β3)2 + (β4)2
π = β9 + 6
π = β25
π = 5
Now,
π πππ =π¦
π= β
4
5 πππ πππ =
π
π¦=
β5
4
πππ π =π₯
π=
β3
5 π πππ =
π
π₯=
β5
3
π‘πππ =π¦
π₯=
4
3 πππ‘π =
3
4
Question No. 9 ππ ππππ½ = βπ
βπ, and terminal side of
the angle is not in quadrant π°π°π°, find the valves of
ππππ½, ππππ½ and ππππππ½.
Solution:
As sin = β1
β2 πππ π‘πππππππ side of angle is not in πΌπΌπΌ
quadrant, so it lies in quadrant πΌπ.
From the figure π¦ = β1 πππ π = β2
By Pythagorean theorem
π2 = π₯2 + π¦2
π₯2 = π2 β π¦2
π₯ = βπ2 β π¦2
π = β(β2)2
β (β1)2
π = β2 β 1
π = β1
π = 1
Now,
ππππ =π¦
π₯= β
1
1= β1
π πππ =π
π₯=
β2
1= β2
πππ πππ =π
π¦=
β2
1= ββ2
Question No.10 If ππππππ½ =ππ
ππ πππ ππππ½ >
π ππππ
The remaining trigonometric functions.
Solution:
As, πππ πππ =13
12 πππ πππ π π πππ ππ + π£π, which is
only possible in quadrant πΌ
From the figure π¦ = 12 πππ π = 13
By Pythagorean theorem
π2 = π₯2 + π¦2
π₯2 = π2 β π¦2
π₯ = βπ2 β π¦2
π = β(13)2 β (12)2
π = β169 β 144
π = β25
π = 5
Now,
π πππ =π¦
π=
12
13 πππ πππ =
π
π¦=
13
12
πππ π =π₯
π=
5
13 π πππ =
π
π₯=
13
5
π‘πππ =π¦
π₯=
12
5 πππ‘π =
5
12
Class 10th Chapter 7 www.notes.pk.com
12 | P a g e
Question No.11 Find the valves of trigonometric
functions at the indicated angles π½ in the right
triangles.
i.
From the figure Hypotenuse = 4 πππ π΅ππ π = 3
By Pythagorean theorem we can find perpendicular.
(ππππ)2 + (π΅ππ π)2 = (π»π¦π. )2
(ππππ. )2 + (3)2 = (4)2 (ππππ)2 = 16 β 9
(ππππ)2 = 7
ππππππππππ’ππ = β7
Now
π πππ =πππ.
π»π¦π.=
β7
4 πππ πππ =
π»π¦π.
πππ.=
4
β7
πππ π =π΅ππ π
π»π¦π.=
3
4 π πππ =
π»π¦π.
π΅ππ π=
4
3
π‘πππ =πππ.
π΅ππ π=
β7
3 πππ‘π =
π΅ππ π
πππ.=
3
β7
ii. From the figure
π»π¦ππππ‘ππππ’π = 17
ππππππππππ’πππ = 8
π΅ππ π = 15
Now
π πππ =πππ.
π»π¦π.=
8
17 πππ πππ =
π»π¦π.
πππ.=
17
8
πππ π =π΅ππ π
π»π¦π.=
15
17 π πππ =
π»π¦π.
π΅ππ π=
15
17
π‘πππ =πππ.
π΅ππ π=
8
15 πππ‘π =
π΅ππ π
πππ.=
15
8
iii. From the figure
βπ¦πππ‘ππππ’π = 7 π΅ππ π = 3
we can find perpendicular by Pythagorean
theorem.
(π΅ππ π)2 + (ππππ)2 = (π»π¦π. )2 (ππππ. )2 + (3)2 = (7)2
(ππππ. )2 = 40 β 9
(ππππ. )2 = 40
ππππ. = β40
ππππ. = β4 Γ 10 Now.
π πππ =πππ.
π»π¦π.=
2β10
7 πππ πππ =
π»π¦π.
πππ.=
2β10
β7
πππ π =π΅ππ π
π»π¦π.=
3
7 π πππ =
π»π¦π.
π΅ππ π=
7
3
π‘πππ =πππ.
π΅ππ π=
2β10
3 πππ‘π =
π΅ππ π
πππ.=
3
2β10
Question No.12 Find the value of the trigonometric
functions. Do not use trigonometric table or
calculator.
Solution:
we know that 2ππ + π = π, π€βπππ π β π
i. π‘ππ30π
30π = 30 π
180 ππππππ =
π
6ππππππ
π‘ππ30π = tanπ
6=
1
β3
ii. π‘ππ330π
π‘ππ330π = tan(360π β 30π)
= π‘ππ2π βπ
6
= tan (βπ
6)
= β tanπ
6
= β1
β3
iii. π ππ 3300
π ππ330π = sec(3600 β 300)
= sec 2π βπ
6
= π ππ βπ
6
= secπ
6
=2
β3
iv. cotπ
4
=1
tanπ4
=1
π‘ππ45π=
1
1= 1
v. cos2π
3
Class 10th Chapter 7 www.notes.pk.com
13 | P a g e
cos(1200) = β1
2
vi. πππ ππ2π
3
πππ ππ2π
3= πππ ππ1200 =
1
sin(1200)=
1
β32
=2
β3
vii. cos(β4500)
cos(β4500) = cos(β3600 β 900)
πππ β 2π βπ
2
= πππ 2(β1)π βπ
2
πππ π
2= 0
viii. tan(β9π)
tan(β9π) = tan(β8π β π)
= tan [2(β4)π β π]
= tan(β8π + (βπ))
= tan(βπ)
= 0
ix. πππ (β5π
6)
= cos (β5π
6)
= β cosπ
6 = β
β3
2
x. sin7π
6
sin7π
6= sin (2π β
5π
6)
= sin [2π + (β5π
6)]
= sin (β5π
6) = sin(β1500) = β
1
2
xi. cot7π
6
cot7π
6= cot [2π + (β
5π
6)]
= cot (β5π
6)
=1
tan (β5π6 )
=1
tan(β1500)=
1
1
β3
= β3
xii. cos 2250
cos(2250) = cos(1800 + 450)
= cos π +π
4
= β cosπ
4= β
1
β2
Exercise 7.4 Things to know:
cos2 π + sin2 π = 1
1 + tan0 π = sec2 π
1 + cot2 π = πππ ππ2π In problem 1 β 6 simplify each expression to a single
trigonometric functions.
1. sin2 π₯
cos2 π₯
Solution:
β΅sin2 π₯
cos2 π₯ = tan2 π₯
2. π‘πππ₯π πππ₯π πππ₯ Solution:
π‘πππ₯π πππ₯π πππ₯ = π‘πππ₯π πππ₯ (1
πππ π₯)
=π πππ₯
πππ π₯sin π₯
1
πππ π₯
=sin2 π₯
cos2 π₯
= tan2 π₯
3. π‘πππ₯
π πππ₯
Solution:
π‘πππ₯
π πππ₯=
π πππ₯πππ π₯
1cos π₯
=π πππ₯
πππ π₯Γ
πππ π₯
1= π πππ₯
4. 1 β cos2 π₯ Solution:
1 β cos2 π₯ = cos2π₯ + sin2 π₯ β cos2 π₯ = sin2 π₯
5. sec2 π₯ β 1 Solution:
sec2 π₯ β 1 = sec2 π₯ β (sec2 β tan2 π₯)
= sec2 π₯ β sec2 π₯ + tan2 π₯
= tan2 π₯
6. sin2 π₯ . cot2 π₯ Solution:
sin2 π₯ . cot2 π₯ = sin2 π₯ .cos2 π₯
sin2 π₯
= cos2 π₯ in problem 7-24 verify the identities.
7. (1 β π πππ)(1 + π πππ) = π
Solution:
π³. π―. πΊ = (π β ππππ½)(π + ππππ½)
= 1 β sin2 π
= cos2 π = π . π». π
8. π πππ+πππ π
πππ π= 1 + π‘πππ
Solution:
πΏ. π». π =π πππ + πππ π
πππ π
=π πππ
πππ π+
πππ π
πππ π
= π‘πππ + 1
Class 10th Chapter 7 www.notes.pk.com
14 | P a g e
= π . π». π
9. (π‘πππ + πππ‘π)π‘πππ = sec2 π Solution:
πΏ. π». π = (π‘πππ + πππ‘π)π‘πππ
= (π πππ
πππ π+
πππ π
π πππ)
π πππ
πππ π
= (sin2 π + cos2 π
π ππππππ π)
π πππ
πππ π
= (1
π ππππππ π)
π πππ
πππ π
=1
cos2 π
= sec2 π
10. (πππ‘π + πππ πππ)(π‘πππ β π πππ) = π πππ βπππ π
Solution:
πΏ. π». π = (πππ‘π + πππ πππ)(π‘πππ β π πππ)
= (πππ π
π πππ+
1
π πππ) (
π πππ
πππ πβ π πππ)
= (πππ π + 1
π πππ) (
π πππ β π ππππππ π
πππ π)
= (1 + πππ π
π πππ) (
π πππ(1 β πππ π)
πππ π)
= (1 + πππ π)(1 β cos π)
πππ π
=1 β cos2 π
πππ π
=1
πππ πβ
cos2 π
πππ π
= π πππ β πππ π
= π . π». π
11. π πππ+πππ π
tan2 πβ1=
cos2 π
π πππβπππ π
Solution:
πΏ. π». π =π πππ + πππ π
tan2 π β 1
=π πππ + πππ π
sin2 πcos2 π
β 1
=π πππ + πππ π
sin2 π β cos2 πcos2 π
=π πππ + πππ π
sin2 π β cos2 πΓ cos2 π
=π πππ + πππ π
(π πππ + πππ π)(π πππ β πππ π)Γ cos2 π
=1
π πππ β πππ πΓ cos2 π
=cos2 π
π πππ β cos π
= π . π». π
12. cos2 π
π πππ+ π πππ = πππ πππ
Solution:
πΏ. π». π =cos2 π
π πππ+ π πππ
=cos2 π + sin2 π
π πππ
=1
π πππ= πππ πππ
13. π πππ β πππ π = π‘ππππ πππ Solution:
πΏ. π». π = π πππ β πππ π
=1
πππ πβ πππ π
=1 β cos2 π
πππ π
=sin2 π
πππ π
=π πππ
πππ πΓ π πππ
= π‘ππππ πππ
14. sin2 π
πππ π+ πππ π = π πππ
Solution:
πΏ. π». π =sin2 π
πππ π+ πππ π
=sin2 π + cos2 π
πππ π
=1
πππ π
= π πππ
15. π‘πππ + πππ‘π = π ππππππ πππ Solution:
πΏ. π». π = π‘πππ + πππ‘π
=π πππ
πππ π+
πππ π
π πππ
=sin2 π + cos2 π
πππ ππ πππ
=1
π ππππππ π
=1
π πππΓ
1
cos π
= π ππππππ πππ
16. (π‘πππ + πππ‘π)(πππ π + π πππ) = π πππ +
πππ πππ
Solution:
πΏ. π». π = (π‘πππ + πππ‘π)(πππ π + π πππ)
= (π πππ
πππ π+
πππ π
π πππ) (πππ π + π πππ)
= (sin2 π + cos2 π
πππ ππ πππ) (πππ π + π πππ)
= (1
πππ ππ πππ) (πππ π + π πππ)
=πππ π
πππ ππ πππ+
π πππ
πππ ππ πππ
=1
π πππ+
1
πππ π
= πππ πππ + π πππ
π . π». π
17. π πππ(π‘πππ + πππ‘π) = π πππ Solution:
Class 10th Chapter 7 www.notes.pk.com
15 | P a g e
πΏ. π». π = π πππ(π‘πππ + πππ‘π)
= sin π (π πππ
πππ π+
πππ π
π πππ)
= (sin2 π + cos2 π
πππ ππ πππ) π πππ
= (1
πππ ππ πππ) π πππ
=1
πππ π
= π πππ
18. 1+πππ π
π πππ+
π πππ
1+πππ π= 2πππ πππ
Solution:
πΏ. π». π =1 + πππ π
π πππ+
π πππ
1 + πππ π
=(1 + πππ π)2 + sin2 π
π πππ(1 + πππ π)
=(1 + 2πππ π + cos2 π + sin2 π)
sinπ (1 + πππ π)
=1 + 2πππ π + 1
π πππ
=2 + 2πππ π
π πππ(1 + πππ π)
=2(1 + πππ π)
π πππ(1 + πππ π)
=2
π πππ
= 2πππ πππ
= π . π». π
19. 1
1βπππ π+
1
1+πππ π= 2πππ ππ2π
Solution:
πΏ. π». π =1
1 β πππ π+
1
1 + πππ π
=1 + πππ π + 1 β πππ π
(1 β πππ π)(1 + πππ π)
=2
1 β cos2 π
=2
sin2 π
= 2πππ ππ2π = π . π». π
20. 1+π πππ
1βπ πππβ
1βπ πππ
1+π πππ= 4π‘ππππ πππ
Solution:
πΏ. π». π =1 + π πππ
1 β π πππβ
1 β π πππ
1 + π πππ
=(1 + sin π)2 β (1 β π πππ)2
(1 β π πππ)(1 + π πππ)
=1 + 2π πππ + sin2 π β 1 + 2π πππ β sin2 π
1 β sin2 π
=4π πππ
cos2 π
=4π πππ
cos πΓ
1
πππ π
= 4π‘ππππ πππ
21. sin3 π = π πππ β π πππ cos2 π
Solution:
π . π». π = π πππ β π πππ cos2 π
= π πππ(1 β cos2 π)
= π πππ(sin2 π)
= sin3 π = πΏ. π». π
22. cos4 π β sin4 π = cos2 π β sin2 π Solution:
πΏ. π». π = cos4 π β sin4 π
= (cos2 π)2 β (sin2 π)2
= (cos2 π β sin2 π)(cos2 π + sin2 π)
= (cos2 π β sin2 π)(1)
= π . π». π
23. β1+πππ π
1βπ πππ=
π πππ
1βπππ π
πΏ. π». π = β1 + πππ π
1 β π πππ
= β(1 + πππ π)(1 β πππ π)
(1 β πππ π)(1 β πππ π)
= β1 β cos2 π
(1 β πππ π)2
= βsin2 π
(1 β πππ π)2
=π πππ
1 β πππ
= π . π». π
24. βπ πππ+1
π πππβ1=
π πππ+1
π‘πππ
Solution:
= βπ πππ + 1
π πππ β 1
= β(π πππ + 1)(π πππ + 1)
(π πππ β 1)(π πππ + 1)
= β(π πππ + 1)2
sec2 π β 1
= β(π πππ + 1)2
tan2 π
π . π». π
Class 10th Chapter 7 www.notes.pk.com
16 | P a g e
Angle of Elevation:
The angle between the horizontal line through eye
and a lie from eye to the object above the horizontal
line is called angle of elevation.
Angle of depression:
The angle between the horizontal line through eye
and a line from eye to the object below the horizontal
line is called angle of depression.
Exercise 7.5 Question No.1 Find the angle of elevation of the
sun if a 6feet man casts a 3.5 feet shadow.
Solution:
From figure we have
π‘πππ =π΄π΅
π΅πΆ
π‘πππ =6
3.5
π‘πππ = 1.714
π = tanβ1(1.7143)
π = 59.74370
π = 59044β²37β²β²
Question No.2 A true casts a 40 meter shadow when
the angle of elevation of the sun is πππ. Find the
height of the tree.
Solution:
From the figure
Height of tree = ππ΄πΆΜ Μ Μ Μ =?
Length of shadow= ππ΅πΆΜ Μ Μ Μ = 40π
Angle of Elevation = π = 250
Angle of fact that
π‘πππ =ππ΄πΆΜ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
π‘πππ =ππ΄πΆΜ Μ Μ Μ
40
ππ΄πΆΜ Μ Μ Μ = 40 Γ π‘ππ250
ππ΄πΆΜ Μ Μ Μ = 18.652π
So, height of tree is 18.652 π
Question No.3. A feet long ladder is learning against
a wall. The bottom of the wall. Find the acute angle
(angle of elevation) the ladder makes with the
ground.
Solution:
from the figure
Length of ladder = π π΄π΅Μ Μ Μ Μ = 20ππππ‘
Distance of ladder from the wall= π π΅πΆΜ Μ Μ Μ = 5 ππππ‘
π΄ππππ ππ ππππ£ππ‘πππ = π =?
Using the fact that
πππ π =π π΅πΆΜ Μ Μ Μ
π π΄π΅Μ Μ Μ Μ
πππ π =5ππ‘.
20ππ‘.
πππ π = 0.25
π = cosβ1 0.25
π = 75.5225
π = 75.50
ππ π = 75030β²
So, angle of elevation is 75030β²
Class 10th Chapter 7 www.notes.pk.com
17 | P a g e
Question No.4 The base of rectangular is 25 feet and
the height of rectangular is 13 feet. Find the angle
that diagram of the rectangular makes with the base.
Solution:
From the figure
Base of rectangular = ππ΅πΆΜ Μ Μ Μ = 25ππππ‘
Height of rectangular = ππ΅πΆΜ Μ Μ Μ = 13ππππ‘
Diagonal π΄πΆΜ Μ Μ Μ ππ π‘ππππ
Angle between diagonal and base= π
Using the fact that
π‘πππ =ππ΅πΆΜ Μ Μ Μ
ππ΄π΅Μ Μ Μ Μ
π‘πππ =13
25
π = tanβ113
25
π = 27.4744
π = 27.470
So, angle between diagonal and base is 27.470
Question No.5 A rocket is launched and climbs at a
constant angle of πππ. Find the altitude of the rocket
after it travels ππππ meter.
Solution:
From the figure
Distance travelled by rocket = ππ΄π΅Μ Μ Μ Μ = 5000π
π΄ππ‘ππ‘π’ππ ππ ππππππ‘ = ππ΄πΆΜ Μ Μ Μ =?
Angle of elevation = π = 800
Using
π πππ =ππ΄πΆΜ Μ Μ Μ
ππ΄π΅Μ Μ Μ Μ
π ππ800 =ππ΄πΆΜ Μ Μ Μ
5000
ππ΄πΆΜ Μ Μ Μ = 5000 Γ π ππ800
ππ΄πΆΜ Μ Μ Μ = 4924.04π
Question No.6 An aero plane pilot flying at an
altitude of 4000m wishes to make an approaches to
an airport at an angle of πππ ππππ πππ plane be
when the pilot begins to descend?
Solution:
From the figure
Altitude of aero plane= ππ΄πΆΜ Μ Μ Μ = 4000π
Distance of plane from airport= ππ΅πΆΜ Μ Μ Μ =?
Angle of depression= 500
As the altimeter angles of parallel lines are equal, so
angle
π = 500 Using the fact that
π‘πππ =ππ΄πΆΜ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
π‘ππ500 =4000π
ππ΅πΆ
ππ΅πΆΜ Μ Μ Μ =4000π
π‘ππ500
ππ΅πΆΜ Μ Μ Μ = 33356.4m
So, the distance of aero plane from airport is 3356.4m
Question No.7 A guy wire (supporting wire) runs
from the middle of a utility pole to ground. The wire
makes an angle of ππ. ππ ππππ πππ ππππππ and
touch the ground 3 meters from the base of the pole.
Find the height of the pole.
Solution:
From the figure
Height of pole= ππΆπ·Μ Μ Μ Μ =?
Distance of wire from the base of the pole
= ππ΅πΆΜ Μ Μ Μ = 3π
Angle of elevation= π = 78.20
As the wire is attached with the pole at its middle
point π΄ so first we find ππ΄πΆΜ Μ Μ Μ
Using the fact that
Class 10th Chapter 7 www.notes.pk.com
18 | P a g e
π‘πππ =ππ΄πΆΜ Μ Μ Μ
ππ΅πΜ Μ Μ Μ
π‘ππ78.2 =ππ΄πΆΜ Μ Μ Μ
3
ππ΄πΆΜ Μ Μ Μ = 3π Γ π‘ππ78.20
ππ΄πΆΜ Μ Μ Μ = 14.36π
So, Height of pole is = ππ·πΆΜ Μ Μ Μ = 2(ππ΄πΆΜ Μ Μ Μ )
= 2 Γ 14.36π
= 28.72π
Question No.8 A road is inclined at an angle π. ππ
suppose that we drive 2 miles up this road starting
from sea level. How high above sea level are we?
Solution:
From the figure
Distance covered on road = ππ΄π΅Μ Μ Μ Μ = 2πππππ
Angle of inclination = π½ = π. ππ
Height from sea level = ππ΄πΆΜ Μ Μ Μ =?
Using the fact that,
π πππ =ππ΄πΆΜ Μ Μ Μ
ππ΄π΅Μ Μ Μ Μ
ππππ. ππ =ππ΄πΆΜ Μ Μ Μ
π
ππ΄πΆΜ Μ Μ Μ = 2 Γ π ππ5.70
ππ΄πΆΜ Μ Μ Μ = 0.199ππππ
Hence , we are at height of 0.199 mile from the sea
level.
Question No.9 A television antenna of 8 feet height
is point on the top of a house. From a point on the
ground the angle of elevations to the top of the
house is πππ
And the angle of elevation to the top of antenna is
ππ. ππ. find the height of the house.
Solution:
From the figure
Distance of point from house ππ΅πΆΜ Μ Μ Μ = π₯
π―πππππ ππ πππππ = ππ΄πΆΜ Μ Μ Μ = β =?
Height of antenna = ππ΄π·Μ Μ Μ Μ = 8ππππ‘
Angle of elevation of top of house = 170
Angle of elevation of top of antenna= 21.80
In right angled βπ΄π΅πΆ
π‘ππ170 =ππ΄πΆΜ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
π‘ππ170 =β
π₯
π₯ =1
π‘ππ170Γ β
π₯ = 3.271 Γ β β (π)
Now in right angle βπ·π΅πΆ
π‘ππ21.8 =ππΆπ·Μ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
π‘ππ21.8 =ππ΄π·Μ Μ Μ Μ + ππ΄πΆΜ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
π‘ππ21.8 =8 + β
π₯
0.40 =8 + β
3.271β ππππ (π)
0.40 Γ 3.271β = 8 + β
1.3084β β β = 8 (1.3084 β 1)β = 8
0.3084β = 8
β =8
0.3084
β =8
0.3084= 25.94ππππ‘
question No.10 from an observation point, the
angles of depression of two boats in line with this
point are found to πππ πππ πππ find the distance
between the two bosts if the point of observation is
ππππ feet high.
Solution:
From the figure
Height of observation point = ππ΄π·Μ Μ Μ Μ = 4000ππππ‘
Distance between boats = ππ΅πΆΜ Μ Μ Μ =?
Angle of depression of points π΅ πππ πΆ πππ
300 πππ 450 respectively from point A.
As the alternate angles of parallel lines are equal, so
πβ π΅ = 300 πππ πβ πΆ = 450
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19 | P a g e
Now in right angled βπ΄πΆπ·
π‘ππ450 =ππ΄π·Μ Μ Μ Μ
ππΆπ·Μ Μ Μ Μ
1 =4000
ππΆπ·Μ Μ Μ Μ
ππΆπ·Μ Μ Μ Μ = 4000ππππ‘
Now in right angled βACD
π‘ππ300 =ππ΄π·Μ Μ Μ Μ
ππ΅π·Μ Μ Μ Μ
1
β3=
4000
ππ΅πΆΜ Μ Μ Μ + ππΆπ·Μ Μ Μ Μ
1
β3=
4000
ππ΅πΆΜ Μ Μ Μ + 4000
ππ΅πΆΜ Μ Μ Μ + 4000 = 4000β3
ππ΅πΆΜ Μ Μ Μ = 4000β3 β 4000
ππ΅πΆΜ Μ Μ Μ = 6928.20 β 4000
ππ΅πΆΜ Μ Μ Μ = 2928.20ππππ‘
So, the distance between boats is 2928.2 feet.
Question No.11 Two ships, which are in lines with the
base of a vertical cliff are 120 meters apart. The angles of
depression from the top of the cliff to the ship are πππ
and πππ as shown in the diagram.
(a) Calculate the distance BC
(b) Calculate the height CD of the cliff.
Solution:
From the figure
Height of cliff = πΆπ·Μ Μ Μ Μ = β =?
Distance= π΅πΆΜ Μ Μ Μ = π₯ =?
Distance between boats = π΄π΅Μ Μ Μ Μ = 120π
Angles of depression from point D to point A and B are
300 πππ 450 respectively.
As the altitude angles of parallel lines are equal, so πβ π΄ =
300 πππ πβ π΅ = 450
In right angled βπ΅πΆπ·
π‘ππ450 =ππΆπ·Μ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
π =β
π₯
π₯ = β β (π)
Now in right angled βπ΄πΆπ·
π‘ππ300 =ππΆπ·Μ Μ Μ Μ
ππ΄πΆΜ Μ Μ Μ
1
β3=
β
ππ΅πΆΜ Μ Μ Μ + ππ΅πΆΜ Μ Μ Μ
1
β3=
β
120 + π₯
120 + π₯ = β3β
120 + β = β3β
120 = β3β β β
120 = β(β3 β 1)
120 = β(1.7321 β 1)
120 = (0.7321β) 120
0.7321= β
β = 163.91π
As π₯ = β , π π
π₯ = 163.91π ππ 164π
Height of cliff= ππΆπ·Μ Μ Μ Μ = 164π
Question No.12 Suppose that we are standing on a
bridge 30 meter above a river watching a log (piece
of wood) floating towards us. If the angle with the
horizontal to the front of the log is ππ. ππ and angle
with the horizontal to the back of the log is πππ,
πππ ππππ ππ πππ πππ?
Solution:
Height of the observer position = πππΆΜ Μ Μ Μ = 30π
Length of log wood= ππ΄π΅Μ Μ Μ Μ = π₯ =?
Angles of depression from point π of the points A and B are
140 and =16.70 respectively
In right angled βππ΅πΆ
π‘ππ16.70 =πππΆΜ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ
0.30 =30
ππ΅πΆΜ Μ Μ Μ
ππ΅πΆΜ Μ Μ Μ =30
0.30
ππ΅πΆΜ Μ Μ Μ = 100π
Now in right angled βππ΄πΆ
π‘ππ140 =πππΆΜ Μ Μ Μ
ππ΄πΆΜ Μ Μ Μ
0.249 =30
ππ΄π΅Μ Μ Μ Μ + ππ΅πΆΜ Μ Μ Μ
0.249 =30
(π₯ + 100)
0.249(π₯ + 100) = 30
π₯ + 100 =30
0.249
π₯ + 100 = 120.482
π₯ = 120.482 β 100
π₯ = 20.482π
So the length of log is 20.48222m.
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Complied by Shumaila Amin