Measurement Of Angles
Class 11th Mathematics RD Sharma Solution
- Find the degree measure corresponding to the following radian measures (Use π =…
- Find the radian measure corresponding to the following degree measures: (i)…
- The difference between the two acute angles of a right-angled triangle is 2π/5…
- One angle of a triangle is 2/3 x grades, and another is 3/2 x degrees while the…
- Find the magnitude, in radians and degrees, of the interior angle of a regular:…
- The angles of a quadrilateral are in A.P., and the greatest angle is 120^0 .…
- The angles of a triangle are in A.P., and the number of degrees in the least…
- The angle in one regular polygon is to that in another as 3:2 and the number of…
- The angles of a triangle are in A.P. such that the greatest is 5 times the…
- The number of sides of two regular polygons is 5:4 and the difference between…
- A railroad curve is to be laid out on a circle. What radius should be used if…
- Find the length which at a distance of 5280 m will subtend an angle of 1’ at…
- A wheel makes 360 revolutions per minute. Through how many radians does it turn…
- Find the angle in radians through which a pendulum swings if its length is 75…
- The radius of a circle is 30 cm. Find the length of an arc of this circle, if…
- A railway train is traveling on a circular curve of 1500 meters radius at the…
- Find the distance from the eye at which a coin of 2 cm diameter if angular diameter of…
- Find the diameter of the Sun in km supposing that it subtends an angle of 32’ at the eye…
- If the arcs of the same length in two circles subtend angles 65^0 and 110^0 at…
- Find the degree measure of the angle subtended at the centre of a circle of…
- If D, G and R denote respectively the number of degrees, grades and radians in an…
- If the angles of a triangle are in A.P., then the measure of one of the angles in…
- The angle between the minute and hour hands of a clock at 8:30 is Mark the correct…
- At 3:40, the hour and minute hands of a clock are inclined at Mark the correct…
- If the arcs of the same length in two circles subtend angles 65° and 110° at the…
- If OP makes 4 revolutions in one second, the angular velocity in radians per second is…
- A circular wire of radius 7 cm is cut and bent again into an arc of a circle of radius…
- The radius of the circle whose arc of length 15 π cm makes an angle of 3π/4 radian at…
Exercise 4.1
Question 1.Find the degree measure corresponding to the following radian measures (Use π =
)
(i) 
(ii) -
(iii) 
c (iv) (-3)c (v) 11c (vi) 1c
Answer:
We know that π rad = 180°⇒1 rad = 180°/ π
(i) Given 
= (36 × 9) °
= 324°
(ii) Given 
= (30 × -5) °
= -(150) °
(iii) Given 
= (36 × 18) °
= 648°
(iv) Given (-3) c
= -(171° 49’ 5.45”)
≈ -(171° 49’ 5”)
(v) Given 11c
= (90 × 7) °
= 630°
(vi) Given 1c
= (57° 16’ 21.81”)
≈ (57° 16’ 22”)
Question 2.
Find the radian measure corresponding to the following degree measures:
(i) 3000 (ii) 350 (iii) -560 (iv)1350 (v) -3000
(vi) 70 30’ (vii) 1250 30’ (viii) -47o 30’
Answer:
We know that 180° = π rad⇒1° = π/ 180 rad
(i) Given 300°
(ii) Given 35°
(iii) Given -56°
(iv) Given 135°
(v) Given -300°
(vi) Given 7° 30’
We know that 30’ = (1/2)°
∴ 7° 30’ = (7 1/2) °
(vii) Given 125° 30’
We know that 30’ = (1/2)°
∴ 125° 30’ = (1251/2) °
(viii) Given -47° 30’
We know that 30’ = (1/2)°
∴ -47° 30’ = -(471/2) °
Question 3.
The difference between the two acute angles of a right-angled triangle is 2π/5 radians. Express the angles in degrees.
Answer:
Given the difference between the two acute angles of a right-angled triangle is 2π/5 radians.
We know that π rad = 180°⇒1 rad = 180°/ π
Given 
= (36 × 2) °
= 72°
Let one acute angle be x° and the other acute angle be 90° - x°.
Then,
⇒ x° - (90° - x°) = 72°
⇒ 2x° - 90° = 72°
⇒ 2x° = 72° + 90°
⇒ 2x° = 162°
⇒ x° = 162°/ 2
∴ x° = 81° and 90° - x° = 90° - 81° = 9°
Question 4.
One angle of a triangle is 
grades, and another is 
degrees while the third is 
radians. Express all the angles in degrees.
Answer:
Given one angle of a triangle is 2x/3 grades and another is 3x/2 degree while the third is πx/75 radians.
We know that
We know that π rad = 180°⇒1 rad = 180°/ π
Given 
We know that the sum of the angles of a triangle is 180°.
⇒ 45 x° = 180° × 10°
⇒ 45 x° = 1800°
⇒ x° = 1800°/ 45°
∴ x = 40°
∴ The angles of the triangle are
Question 5.
Find the magnitude, in radians and degrees, of the interior angle of a regular:
(i) Pentagon (ii) Octagon (iii) Heptagon (iv) Duodecagon.
Answer:
We know that the sum of the interior angles of a polygon = (n – 2) π
And each angle of polygon 
(i) Pentagon
Number of sides in pentagon = 5
Sum of interior angles of pentagon = (5 – 2) π = 3π
∴ Each angle of pentagon 
(ii) Octagon
Number of sides in octagon = 8
Sum of interior angles of octagon = (8 – 2) π = 6π
∴ Each angle of octagon 
(iii) Heptagon
Number of sides in heptagon = 7
Sum of interior angles of heptagon = (7 – 2) π = 5π
∴ Each angle of heptagon 
(iv) Duodecagon
Number of sides in duodecagon = 12
Sum of interior angles of duodecagon = (12 – 2) π = 10π
∴ Each angle of duodecagon 
Question 6.
The angles of a quadrilateral are in A.P., and the greatest angle is 1200. Express the angles in radians.
Answer:
Let the angles of quadrilateral be (a – 3d) °, (a – d) °, (a + d) ° and (a + 3d) °.
We know that the sum of angles of a quadrilateral is 360°.
⇒ a – 3d + a – d + a + d + a + 3d = 360°
⇒ 4a = 360°
∴ a = 90°
Given the greatest angle = 120°
⇒ a + 3d = 120°
⇒ 90° + 3d = 120°
⇒ 3d = 120° - 90°
⇒ 3d = 30°
⇒ d = 10°
Hence, the angles are:
⇒ (a – 3d) ° = 90° - 30° = 60°
⇒ (a – d) ° = 90° - 10° = 80°
⇒ (a + d) ° = 90° + 10° = 100°
⇒ (a + 3d) ° = 120°
Angles of quadrilateral in radians:
Question 7.
The angles of a triangle are in A.P., and the number of degrees in the least angle is to the number of degrees in the mean angle as 1:120. Find the angle in radians.
Answer:
Let the angles of the triangle be (a – d) °, a° and (a + d) °.
We know that the sum of the angles of a triangle is 180°.
⇒ a – d + a + a + d = 180°
⇒ 3a = 180°
∴ a = 60°
Given 
⇒ 120 – 2d = 1
⇒ 2d = 119
∴ d = 59.5
Hence, angles are:
⇒ (a – d) ° = 60° – 59.5° = 0.5°
⇒ a° = 60°
⇒ (a + d) ° = 60° + 59.5° = 119.5°
∴ Angles of triangle in radians:
Question 8.
The angle in one regular polygon is to that in another as 3:2 and the number of sides in first is twice that in the second. Determine the number of sides of two polygons.
Answer:
Let the number of sides in the first polygon be 2x and the number of sides in the second polygon be x.
We know that angle of an n-sided regular polygon 
⇒ The angle of the first polygon 
⇒ The angle of the second polygon 
Thus,
⇒ 2x – 2 = 3x – 6
∴ x = 4
Thus,
Number of sides in the first polygon = 2x = 8
Number of sides in the second polygon = x = 4
Question 9.
The angles of a triangle are in A.P. such that the greatest is 5 times the least. Find the angles in radians.
Answer:
Let the angles of the triangle be (a – d) °, a° and (a + d) °.
We know that the sum of angles of triangle is 180°.
⇒ a – d + a + a + d = 180°
⇒ 3a = 180°
∴ a = 60°
Given greatest angle = 5 × least angle
⇒ 60 + d = 300 – 5d
⇒ 6d = 240
∴ d = 40
Hence, angles are:
⇒ (a – d) ° = 60° – 40° = 20°
⇒ a° = 60°
⇒ (a + d) ° = 60° + 40° = 100°
∴ Angles of triangle in radians:
Question 10.
The number of sides of two regular polygons is 5:4 and the difference between their angle is 90. Find the number of sides of the polygons.
Answer:
Let the number of sides in the first polygon be 5x and the number of sides in the second polygon be 4x.
We know that angle of an n-sided regular polygon
⇒ The angle of the first polygon 
⇒ The angle of the second polygon 
Thus,
⇒ 
∴ x = 2
Thus,
Number of sides in the first polygon = 5x = 10
Number of sides in the second polygon = 4x = 8
Question 11.
A railroad curve is to be laid out on a circle. What radius should be used if the track is to change direction by 250 at a distance of 40 meters?
Answer:
Given length of arc = 40 m
And θ = 25°
We know that 180° = π rad⇒1° = π/ 180 rad
We know that 
= 91.64 m
So, the radius of the track should be 91.64 m.
Question 12.
Find the length which at a distance of 5280 m will subtend an angle of 1’ at the eye.
Answer:
Given radius = 5280 m
We know that 
And know that
= 1.5365 m
Question 13.
A wheel makes 360 revolutions per minute. Through how many radians does it turn in 1 second?
Answer:
Given the number of revolutions taken by the wheel in 1 minute = 360
Number of revolution taken by the wheel in 1 second = 360/ 6 = 6
We know that 1 revolution = 2π radians
∴ Number of radians the wheel will turn in 1 second = 6 × 2π = 12π
Question 14.
Find the angle in radians through which a pendulum swings if its length is 75 cm and the tip describes an arc of length (i) 10 cm (ii) 15 cm (iii) 21 cm.
Answer:
Given radius = 75 cm
We know that
(i) Given length of arc = 10 cm
(ii) Given length of arc = 15 cm
(iii) Given length of arc = 21 cm
Question 15.
The radius of a circle is 30 cm. Find the length of an arc of this circle, if the length of the chord of the arc is 30 cm.
Answer:
Let AB be chord and O be the centre if the circle.
Here AO = BO = AB = 30 cm
∴ ΔAOB is an equilateral triangle.
Given radius = 30 cm
And θ = 60°
We know that 180° = π rad⇒1° = π/ 180 rad
We know that
= 10π cm
Question 16.
A railway train is traveling on a circular curve of 1500 meters radius at the rate of 66 km/hr. Through what angle has it turned in 10 seconds?
Answer:
Given time is 10 seconds.
And speed = 66 km/h 
We know that
Now radius of curve = 1500 m
We know that
So, the train will turn 11/ 90 radian in 10 seconds.
Question 17.
Find the distance from the eye at which a coin of 2 cm diameter should be held so as to conceal the full moon whose angular diameter is 31’.
Answer:
Let PQ be the diameter of the coin and E be the eye of the observer.
Let the coin be kept at a distance r from the eye of the observer to hide the moon completely.
We know that
= 221.7 cm
Question 18.
Find the diameter of the Sun in km supposing that it subtends an angle of 32’ at the eye of an observer. Given that the distance of the Sun is 91 x 106 km.
Answer:
Let PQ be the diameter of the sun and E be the eye of the observer.
The distance between the Sun and Earth is large, so we will take PQ as arc PQ.
Given radius = 91 × 106 km
We know that
= 847407.4 km
Question 19.
If the arcs of the same length in two circles subtend angles 650 and 1100 at the center, find the ratio of their radii.
Answer:
Let the angles subtended at the centers by the arcs and radii of first and second circles be θ1 and r1 and θ2 and r2.
Then,
We know that 180° = π rad⇒1° = π/ 180 rad
We know that
Question 20.
Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use π = 
).
Answer:
Given length of arc = 22 cm
And radius = 100 cm
We know that
∴ The angle subtended at the centre by the arc:
= 12° 36’.
Mcq
Question 1.Mark the correct alternative in the following:
If D, G and R denote respectively the number of degrees, grades and radians in an angle, then
A. 
B. 
C. 
D. 
Answer:
Let θ be the angle which is measure in degree, radian and grade
We know that 90°=1 right angle
right angle
right angles
right angle ……(1)
Also we know that ,π radians=2 right angles
right angle
right angles
right angles ……(2)
Also we know that, 100 grades=1 right angle
⇒1 grade
right angle
⇒G grade
right angles
right angles ……(3)
From (1),(2) and (3)
Question 2.
Mark the correct alternative in the following:
If the angles of a triangle are in A.P., then the measure of one of the angles in radians is
A. 
B. 
C. 
D. 
Answer:
Here, angles of triangle are in A.P.
So, Let angles of triangle are a,a+d,a+2d.
We know that, sum of angles of triangle is π.
∴ a+a+d+a+2d=π
∴ 3a+3d=π
∴ 3(a + d) =π
Also, by our assumption, a + d is one angle of triangle.
So, required measure of one of the angles is 
.
Question 3.
Mark the correct alternative in the following:
The angle between the minute and hour hands of a clock at 8:30 is
A. 80°
B. 75°
C. 60°
D. 105°
Answer:
We know, in clock 1 rotation gives 360°
i.e. 60 minutes=360° and 12 hours=360°
So,1 minute=6° and 1 hour=30°
Now, For hour hand:
8 hours=8×30°=240° and for another 30 minute (which is half of hour) =30°÷2=15°
i.e. angle traced by hour hand is 240°+15°=255°
Now, For minute hand:
30 minute=30×6°=180°
i.e. angle traced by minute hand is 180°.
So, the angle between hour hand and minute hand=255°-180°
=75°
Question 4.
Mark the correct alternative in the following:
At 3:40, the hour and minute hands of a clock are inclined at
A. 
B. 
C. 
D. 
Answer:
We know, in clock 1 rotation gives 360°
i.e. 60 minutes=360° and 12 hours=360°
So,1 minute=6° and 1 hour=30°
Now, For hour hand:
3 hours=3×30°=90° and for another 40 minute= (30°÷60)×40=20°
i.e. angle traced by hour hand is 90°+20°=110°
Now, for minute hand:
40 minute=40×6°=240°
i.e. angle traced by minute hand is 240°.
So, the angle between hour hand and minute hand=240°-110°
=130°
Question 5.
Mark the correct alternative in the following:
If the arcs of the same length in two circles subtend angles 65° and 110° at the centre, then the ratio of the radii of the circles is
A. 22 : 13
B. 11 : 13
C. 22 : 15
D. 21 : 13
Answer:
Let radius of two circles be the r1 and r2
Let θ1 and θ2 be the subtend angles of arcs of two circles
i.e. θ1=65° and θ2=110°
We know that arc length,
l = r × θ
Here, arc lengths of two circles are same.
∴ r1×θ1=r2×θ2
∴r1 :r2 =22:13
Question 6.
Mark the correct alternative in the following:
If OP makes 4 revolutions in one second, the angular velocity in radians per second is
A. π
B. 2 π
C. 4 π
D. 8 π
Answer:
We know that, 1 revolution =2×π radians
Now, Angular velocity
=8×π
Question 7.
Mark the correct alternative in the following:
A circular wire of radius 7 cm is cut and bent again into an arc of a circle of radius 12 cm. The angle subtended by the arc at the centre is
A. 50°
B. 210°
C. 100°
D. 60°
E. 195°
Answer:
Here, radius of circular wire is r=7 cm
So, length of wire=2×π×r
=2×π×7
=14×π
Wire is cut and bent again into an arc of a circle of radius 12 cm.
So, length of arc=length of wire=14×π
We know, angle subtended by the arc is given by,
=210°
Question 8.
Mark the correct alternative in the following:
The radius of the circle whose arc of length 15 π cm makes an angle of 3π/4 radian at the centre is
A. 10 cm
B. 20 cm
C. 
cm
D. 
cm
Answer:
Here, arc length l=15π cm
Angle 
We know, angle subtended by the arc is given by,
cm