Question 1: Find the degree corresponding to the following radian measures (Use ):

i) ii) iii) iv) v) vi)

Answer:

i)

ii)

iii)

iv)

v)

vi)

Question 2: Find the radian measure corresponding to the following degree measures:

i) ii) iii) iv) v) vi) vi) vii)

Answer:

i)

ii)

iii)

iv)

v)

vi)

vii)

viii)

Question 3: The difference between the two acute angles of a right angled triangle is radians. Express the angles in degrees.

Answer:

Let and be two acute angles of a right angles triangle.

… … … … … i)

Also in a right angled triangle

… … … … … ii)

Adding i) and ii) we get

Hence and

Question 4: One angle of a triangle is grades and another is degrees while the third angle is radians. Express all the angles in degrees.

Answer: Let the three angles be and

gradiant

In the triangle

Question 5: Find the magnitude, in radians and degrees, of interior angle of: i) pentagon ii) octagon iii) heptagon iv) duodecagon

Answer:

Interior angles of a polygon with sides

i) Pentagon

Interior angle

Therefore in radians, interior angle

ii) Octagon

Interior angle

Therefore in radians, interior angle

iii) Heptagon

Interior angle

Therefore in radians, interior angle

iv) Duodecagon

Interior angle

Therefore in radians, interior angle

Question 6: The angles of a quadrilateral are in A.P. and the greatest angles is . Express the angles in radians.

Answer:

Let the angles of quadrilateral in degrees be

We know, sum of angles in quadrilateral

Largest angle

angles are and

In radians angles are , and

, and

Question 7: The angles of a quadrilateral are in A.P. and the number of degrees in the least angles is to the number of degrees in the mean angle as . Find the angles in radians.

Answer:

Let and be the three angles

It is given that and are in an AP

Let , and

Given that

in radians

and

and

Question 8: The angle in a regular polygon is to that in another is and the number of sides in first is twice that in second. Determine the number of sides of two polygons.

Answer:

Let and be the number of sides of two regular polygons respectively.

… … … … … i)

Also given

Substituting in i) we get

Hence and

Question 9: The angles of a triangle are in A.P. such that the greatest angle is five times the least. Find the angles in radians.

Answer:

Let and be the three angles

Let and

Given

In radians

Question 10: The number of sides of two regular polygons are as and the difference between their angles is 9^o. Find the number of sides of the polygon.

Answer:

Let and be the number of sides of two regular polygons respectively.

Also given

Substituting

Hence and

Question 11: A rail road curve is to be laid out on a circle. What radius should be used if the track is to change direction by in a distance of meters.

Answer:

Let be the rail road

We know that

meters

meters

Question 12: Find the length which is at a distance m will subtend an angle of at the eye.

Answer:

Given Radius meters

We know that

meters

Question 13: A wheel makes revolutions in a minute. Through how many radians does it turn in second.

Answer:

Given revolutions per minute revolution per second

In revolution, wheel will cover

In one second the wheel will cover

In radians,

Question 14: Find the angle in radians through which a pendulum swings if its length is cm and the tip describes as arc of length i) cm ii) cm iii) cm

Answer:

i) Let length of pendulum cm m

cm m

ii) m

cm m

iii) m

cm cm

Question 15: The radius of a circle is cm. Find the length of an arc of this circle, if the length of the chord of the arc is cm.

Answer:

radius of circle cm m

chord cm m

is equilateral triangle.

Let

m cm

Question 16: A railway train is travelling on a circular curve of m radius at a rate of km/hr. Through what angle has it turned in seconds?

Answer:

radius m

The train turns in seconds

Speed of train km/hr m/s m/s

In seconds, train travels m

m

the train will turn by in seconds

Question 17: Find the distance from the eye at which a coin of cm diameter should be held so as to conceal the full moon whose angular diameter is .

Answer:

Let be the distance at which coin needs to be placed to completely conceal the full moon.

Given cm cm

m

Therefore the coin should be placed m away from the eye.

Question 18: Find the diameter of the sun in km supposing that it subtends an angle of at the eye of an observer. Given that the distance of the sun is km.

Answer:

Given

Since

km

Therefore Distance of sun is km

Question 19: If the arcs of the same length in two circles subtend angles and at the center, find the ratio of their radii.

Answer:

Let and be two circles with same arc length

If and are two angles subtended by the arcs on respective circles

and

We know,

Similarly,

Therefore

Question 20: Find the degree measure of the angle subtended at the center of the circle of radius cm by an arc of length cm. (Use ):

Answer:

Let cm

cm

Let be the angle subtended by the at center