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**1, Find the quadrants in which the following angles lie :**

i) 45^{o} ii)120^{o}

i) 45^{o} lies in the first quadrant

ii) 120^{o} lies in the second quadrant

**2, Find the quadrants in which the following angles lie:**

I, 200^{o} ii,300^{o}

i) 200^{o} lies in the third quadrant

ii) 300^{o} lies in the fourth quadrant

**3, Find the quadrants in which the following angles lie.**

i, 500

Ans: 500^{o} = 360^{o} x 1 + 90^{o} +50^{o}

**4, Find the quadrants in which the following angle lie.**

i) 600^{o}

600^{o} = 360^{o} x 1 + 180^{o} + 60^{o}

**5, Find the quadrants in which the following angles lie.**

i) 800^{o}

800^{o} = 360^{o} x 2 + 80^{o}

**6, Find the quadrants in which the following angle lie.**

i) -60^{o}

**7, Find the quadrants in which the following angle lies.**

i) -120^{o}

**8, Find the quadrants in which the following angle lie:**

i) -1050^{o}

-1050^{o} = 2 x ( -360^{o} ) + ( -330^{o})

**9,Find the radian measure corresponding to the following degree measure.**

i) 25^{o}

We know that , 180^{ o} = π c

Hence, radian measure of 25^{o} is

**10, Find the radian measure corresponding to the following degree measure.**

i) -47^{o} 30’

**11,Find the radian measure corresponding to the following degree measure.**

i) 240^{o}

We know that 180 ^{o} = π c

Hence, radian measure of 240^{o} is **12, Find the radian measure corresponding to the following degree measure.**

i) 520^{o}

Hence, radian measure of 520 ^{o} is

**13, Find the radian measure corresponding to the following degree measure.**

i) ^{5π}/3

We know that 180 ^{o} = π c

= 300^{o}

Hence, radian measure of 5π/3 radian os 300^{o}

**14, Find the degree measure corresponding to the following radian measure.**

i) ^{11}/16

**15, Find the degree measure corresponding to the following radian measure.of -4 ^{c}**

= - 229

Hence degree measure of -4

**16, Express 5 ^{o} 37’ 30” into radians.**

**17, The angles of a triangle are in A.P. The ratio of number of degrees of the least angle, is to the number of radians in the gretest angle is given as 60: π. Find the angles in degrees.**

Let the three angles in A.P be a –d, a a +d . Since, the sum of angles of a triangle = 180^{o}

⇒ a –d, a a + d+ a = 60^{o} ⇒

∴ The angles are ( 60 - d ) o, 60^{o}, ( 60 + d )^{o}.

Again the number of degree in the least angle = ( 60 – d ) ^{o}.

And greatest angle = ( 60 – d ) ^{o} = ( 60 + d ) ^{o} ^{π}/ 80 radians

∴ By the given condition ,

**18, The circular measure of two angles of a triangle are ½ radian and 1/3, radian what is the number of degrees in the third angle?**

We know that the sum of angles of a triangle is 180^{o} i.e. π radians. two angles are ½ radian and ^{1}/3 radians.

**19, find the angle between the minute hand and the hour – hand of a clock at**

i) 4 : 30 P.M.

We know that the hour-hand completes one rotation in 12 hours whereas the minute – hand completes one rotation in one hour.

i) At 4 : 30 PM., let the hour hand and the minute – hand be along OB and OC respectively.

∴ Required angle = ∠ BOC

= ( Angle subtended by the minute hand in 30 minutes) - (angle subtended by the hour – hand in 4 hours 30 minutes i.e.^{ 9}/2 hours)

**20, Find the angle between the minute hand and the hour – hand of a clock at :**

At 10: 40 PM., let the hour – hand and the minute – hand be along OC and OB respectively.

Required angle = ∠ BOC

= ( angle subtended by the hour – hand in 1o hours 40 minutes i.e. 10 ^{2}/3 hours ) – ( angle subtended by the minute – hand in 40 minutes )

**21, What is positive angle ?**

If the ray rotates OB about 0 from, the initial position OA in the describes a positive angle. In the anticlockwise direction, we say that it describes a positive angle. In the figure,∠ AOB is a positive angle ,

For example,

( i ) ∠ AOB = + 60^{o}, ( ii ) ∠ xoy = + 135^{o}

**22, what is negative angle ?**

If the ray rotates about 0 from the position OA in the clock wise direction we say that it describes a negative angle. In the figure,∠ AOB is a negative angle.

for example :

( i ) ∠ AOB = - 30^{o}, ( ii ) ∠ xoy = - 120^{o},

**23, Define degree measure.**

In this system, the unit of measuring an angle is a degree. If a right angle is divided in 90 equal parts, then each part is called a degree. One is divided in 60 equal parts, each part is called a minute. One minute is further divided in 60 equal parts, each part is called a second.

1 right angle = 90 degrees ( written as 90^{o})

1 degree = 60 minutes ( written as 60’

1 minute = 60 seconds ( written s 60” )**24,** **Define co – terminal Angles:**

Two angles with different measures, but having the same initial sides and the same terminal sides are known as co – terminal angles.

eg: (i) The angles 60^{o} and – 300^{o} are co terminal

( ii) The angles, 240^{o} and -120^{o} are co terminal

**25, Define Radian Measure.**

In this system the angle subtended the arc at the centre of a circle is measure by dividing the length of the arc by the length of the radius.

In this sytem, the angle is measured in radians.

**26, Define Radian.**

Radian is the angle subtended at the centre of a circle by an arc, whose length is equal to the radius.

**27, Prove that Radian is a constant Angle.**

Let 0 be the centre of a circle of radius r

Now, we have to prove that ∠ AOB is constant. Extend AO to meet the circumference at C. Since Arc subtends∠ AOB at the centre and arc subtends straight angle AOC.

**28, Express in circular measures and also in degrees the angle of a regular octagon.**

A regular octagon has 8 equal sides

The sum of the 8 exterior angle = 360^{o}

∴ Each exterior angle = 360^{o}/8 = 45^{o}

∴ Each interior angle = 180^{o} - 45^{o} = 135^{o}

Now, we know that 180^{o} = π radian

Hence , the angle of a regular octagon is = 135^{o} = ^{3π}/4 radians.

**29,Express in circular measure and also in degree the angle of a regular polygon of 40 sides.**

Each exterior angle of a regular polygon of 40 sides = 360^{o}/40 = 9^{o}

Each interior angle = 180^{o} – 9^{o} = 171^{o}

Now , we know that, 180o = π radians

∴ 171^{o} = ^{π} /180 x 171 radians = ^{19π} / 20 radians

Hence the angle of a regular polygon of 40 sides is ^{19π} / 20.

**30. Prove that the number of radians in an angle subtended by an arc of a circle at the centre = arc/ radius. Ie) θc= 1/r, **

Where θc is the angle subtended at the centre of a circle of radius r by an arc of length l.

Let us consider a circle with centre 0 and radius r. Let =∠ AOB θcand arc AB = l.

Let C be a point on arc AB, Such that arc ac = r

Then ∠ AOB = 1 c

We know from geometry, that the angles at the centre of a circle are proportional to the arcs subtending them .

**31, Find the length of an arc of a circle radius 3 cm, if the angle subtended at centre is 30o (π= 3.14 )**

Let l be the length of the arc

We know that, Angle θ = l/r

Hence, the arc length is 1.57 cm.

**32,In a circle of diameter 40 cm the length of a chord is 20 cm. Find the length of the minor arc of the chord.**

We have, Radius ( r ) =^{ 40}/2 = 20 cm

And length of chord = 20 cm

⇒ Δ OAB is an equilateral triangle

**33, Find the angle in radian through which a pendulum swings and its length is 74 cm and the tip describes an arc o****f length 21cm.**

Length of rope = 75 cm

⇒ Radius = 75

Length of the arc = 21 cm

and, we know that angle = ^{Arc}/ radius

=^{21}/ 75 radian

=^{7}/25 radians

**34, Express 104o 36' in the circular system**

**35. The angles of a triangle are in the ratio 2 : 3: 4. Express the angles in circular measwure as well as in degrees.**

Let the angles of the triangle be ( 2 x )^{o}, ( 3 x)^{o}, ( 4x) ^{o}

∴ ( 2 x )^{o}, ( 3 x )^{o}, ( 4x) ^{o} = 180^{o}

⇒ 9x = 180^{o}

⇒ x = 20

∴ The angles are 40^{o}, 60^{o}, 80^{o}

Now 180 = π radians implies

**36, Find in radius the angle of a regular octagon**

No. of sides = 8

No. of exterior angles = 8

sum of all exterior angles = 360^{o}

Each exterior angle = 360^{o}/8 = 45^{o}

Each interior angle = 180^{o} – 45^{o}

= 135^{o} = ( 135 x ^{π}/180 ) radians

= ^{3π}./4 radians.

**37, If in a circle of radius r, an arc of length l subtends an angle θ radians, then prove that θ = l/ r**

Let o be the centre of the circle of radius r, let arc AB = l and arc AC= r, where units of l and r are same.

Since, angles at the centre of a circle are proportional to the arcs on which they stand, we have

**38, Find the length of an arc of a circle of radius 10 am which subtends an angle of 45 ^{o} at the centre.**

Let ∠ AOB= 45

arc AB = l cm

**39, A railway train is travelling on a curve of 500 m radius at the rate of 66 kph. Through what angle will it turn in 20 seconds ?**

**40, In a circle of diameter 40 cm, the length of u chord is 20 cm. Find the length of the minor arc of the chord.**

Let o be the centre of the circle of diameter 40 cm.

Let BC be the chord of length 20 cm

**41, If the arcs of the same length in two circles subtend angles 75 ^{o} and 120^{o} at the centre, Find the ratio of their raddii.**

Let r

Let l be the length of each arc.

**42. Express angles 60 ^{o}, 75^{o}, 115^{o}, in the circular system.**

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