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Trigonometry: Concept Of Measurement Of Angle

Class 10th Mathematics West Bengal Board Solution
Let Us Work Out 20
  1. Let us express the following into degrees, minutes and seconds. (i) 832’ (ii) 6312’’…
  2. Let us determine the circular values of the followings (i) 60° (ii) 135° (ii) -150°…
  3. In ΔABC, AC=BC and BC is extended upto the point D. If ∠ACD=144°, then let us determine…
  4. If the difference of two acute angles of a right-angled triangle is 2 pi /5 , then let…
  5. The measure of one angle of a triangle is 65° and other angle is pi /12 ; let us write…
  6. If the sum of two angles is 135° and their difference is pi /12 ; then let us determine…
  7. If the ratio of three angles of a triangle is 2:3:4, then let us determine the circular…
  8. The length of a radius of a circle is 28 cm. Let us determine the circular value of…
  9. The ratio of two angles subtended by two arcs of unequal lengths at the centre is 5:2…
  10. A rotating ray makes an angle -5 1/12 π. Let us write by calculating, in which…
  11. I have drawn an isosceles triangle ABC whose included angle of two equal sides is…
  12. The base BC of the equilateral triangle ABC is extended upto the point E so that…
  13. If the measures of three angles of quadrilateral are pi /3 , 5 pi /6 and 90°…
  14. Q14A1 The end point of the minute hand of a clock rotates in 1 hourA. pi /4 radian B. pi…
  15. Q14A2 pi /6 radian equals toA. 60° B. 45° C. 90° D. 30°
  16. Q14A3 The circular value of each internal angle of a regular hexagon isA. pi /3 B. 2 pi /3…
  17. Q14A4 The measurement of Θ in the relations to S=rΘ is determined by A. sexagesimal system…
  18. Q14A5 In cyclic quardrilateral ABCD, if ∠a=120°, then the circular of ∠C isA. pi /3 B. pi…
  19. Let us write whether the following statements are true or false: (i) The angle,…
  20. Let us fill in the blanks: (i) π radian is a ______angle. (ii) In sexagesimal system…
  21. Q15A1 If the value of an angle in degree is D and in radian is R; then let us determine…
  22. Let us write the value of complementary angle of the measure 63°35’15’’…
  23. If the measures of two angles of a triangle are 65°56’55’’ and 64°3’5’’, then let us…
  24. In a circle, if an arc of 220 cm. length subtends an angle of measure 63° at the…
  25. Let us write the circular value of an angle formed by the end point of hour hand of a…
Let Us Work Out 20
Question 1.

Let us express the following into degrees, minutes and seconds.

(i) 832’ (ii) 6312’’

(iii) 375’’ (iv)

(v) 72.04°
Answer:

(i) 832’

1 degree = 60 minutes





(ii) 6312’’


1 degree = 3600 seconds






⇒ 6312’’ = 1 degree 45 minutes 12 seconds


(iii) 375’’


1 degree = 3600 seconds





⇒ 375’’ = 6 minutes 15 seconds
Question 2.

Let us determine the circular values of the followings

(i) 60° (ii) 135°

(ii) -150° (iv) 72°

(v) 22°30’ (vi) -62°30’

(vii) 52°52’30’’
Answer:

(i) 60°

∵ 180° = π




(ii) 135°


∵ 180° = π




(ii) -150°


∵ 180° = π




(iv) 72°


∵ 180° = π




(v) 22°30’


⇒ 20° + 30’





∵ 180° = π




(vi) -62°30’


⇒ -60° - 30’






(vii) 52°52’30’’


⇒ 52° + 52’ + 30’’






∵ 180° = π
Question 3.

In ΔABC, AC=BC and BC is extended upto the point D. If ∠ACD=144°, then let us determine the circular value of each of the angles of ΔABC.
Answer:


∠ACD = 144°


∵ DCB is straight line, ∠DCB = 180°


⇒ ∠ACB = 180° - 144° = 36°


∵ AC=BC, by opposite angle property


⇒ ∠CAB = ∠CBA = x


By Angle sum Property


⇒ ∠CAB + ∠CBA + ∠ACB = 180°


⇒ 2x + 36° = 180°


⇒ x = 72°


⇒ ∠CAB = ∠CBA = 72° and ∠ACB = 36°




And ∠ACB = 36°
Question 4.

If the difference of two acute angles of a right-angled triangle is , then let us write the sexagesimal values of two angles.
Answer:

let the two angles be x and y.

∵ it is a right-angled triangle


…[1]


Also, by question


…[2]


Adding eq. [1] and eq. [2]


⇒


…[3]


By eq. [1] and eq. [3]



Converting x to sexagesimal angle



⇒



⇒ x° = 81°


∵ y° = 90° – x°


⇒ y° = 9°
Question 5.

The measure of one angle of a triangle is 65° and other angle is ; let us write the sexagesimal value and circular value of third angle.
Answer:

converting to sexagesimal


= 15°


Let c be the third angle


∵ sum of angles of a triangle = 180°


⇒ 65° + 15° + c = 180°


⇒ c = 100°


Circular value of c
Question 6.

If the sum of two angles is 135° and their difference is ; then let us determine the sexagesimal value and circular value of two angles.
Answer:

converting into sexagesimal value


⇒ 15°


Let the two angles be x and y.


x + y = 135° …[1]


x - y = 15° …[2]


adding eq. [1] and eq. [2]


⇒ 2x = 150°


⇒ x = 75° …[3]


By [1] and [3]


y = 60°


converting x to circular




converting y to circular
Question 7.

If the ratio of three angles of a triangle is 2:3:4, then let us determine the circular value of the greatest angle.
Answer:

let the angles be 2x, 3x, 4x

∵ sum of angles of a triangle = 180°


⇒ 2x + 3x + 4x = 180°


⇒ 9x = 180°


⇒ x = 20°


Angles of the triangle


2x = 40°


3x = 60°


4x = 80°


Circular value of 80°
Question 8.

The length of a radius of a circle is 28 cm. Let us determine the circular value of angle subtended by an arc of 5.5 cm length at the centre of this circle.
Answer:

let θ be the angle subtended by the arc.

length of arc = rθ


⇒ 28×θ = 5.5
Question 9.

The ratio of two angles subtended by two arcs of unequal lengths at the centre is 5:2 and if the sexagesimal value of the second angle is 30°. Then let us determine the sexagesimal value and the circular value of the first angle.
Answer:

let the length of arcs be 5x and 2x

Let r be the radius of the circle


…[1]


Let θ be the angle subtended by the arc of length 5x.


…[2]


By dividing eq. [1] and eq. [2]





⇒ θ = 75°
Question 10.

A rotating ray makes an angle π. Let us write by calculating, in which direction the ray has completely rotate and there after what more angle it has produced.
Answer:

angle of the ray =

The negative sign shows that ray has rotated clockwise.


Adding multiples of 2π




∵ it is greater than , so it is in 2nd quadrant.
Question 11.

I have drawn an isosceles triangle ABC whose included angle of two equal sides is ∠ABC=45°; the bisector of ∠ABC intersects the side AC at the point D let us determine the circular values of ∠ABD, ∠BAD, ∠CBD and ∠BCD.
Answer:


∠ABC = 45°




∵ BD is the angle bisector of ∠ABC



∠BAD + ∠ABC+ ∠BCD = π


∵ ABC is an isosceles triangle


⇒ ∠BAD = ∠BCD = x
Question 12.

The base BC of the equilateral triangle ABC is extended upto the point E so that CE=BC. By joining A,E, let us determine the circular values of the angles of ΔABC
Answer:


∠ABC = ∠BAC = ∠BCA = 60°




∠ACE + ∠ACB = 180°


⇒ ∠ACE = 180° - 60°


⇒ ∠ACE = 120°


∵ BC = CE and BC = AC


⇒ AC = AE


⇒ ∠CAE = ∠AEC = x


∠CAE + ∠AEC + ∠ACE = 180°


⇒ 2x + 120° = 180°


⇒ x = 30°





And ∠ACE = 120°
Question 13.

If the measures of three angles of quadrilateral are and 90° respectively, then let us determine and write the sexagesimal and circular values of fourth angle.
Answer:

sum of angles of quadrilateral = 2π

Let the fourth angle be x






⇒ x = 60°
Question 14.

The end point of the minute hand of a clock rotates in 1 hour
A. radian

B. radian

C. π radian

D. 2π radian
Answer:

angle of complete circle = 2π


Minute hand completes 1 circle in an hour.
Question 15.

radian equals to
A. 60°

B. 45°

C. 90°

D. 30°
Answer:


⇒ 30°
Question 16.

The circular value of each internal angle of a regular hexagon is
A.

B.

C.

D.


Answer:

Sum of internal angle of a polygon = 180(n-2)


⇒ internal angle of a regular polygon=


For hexagon n = 6




⇒ 120°
Question 17.

The measurement of Θ in the relations to S=rΘ is determined by

A. sexagesimal system

B. circular system

C. Those two methods

D. None of these
Answer:

Circumference of a circle is 2πr


Where 2π is the angle subtended in circular system and r is the radius.
Question 18.

In cyclic quardrilateral ABCD, if ∠a=120°, then the circular of ∠C is
A.

B.

C.

D.


Answer:

The sum of opposite angle in a cyclic quadrilateral = π


Converting 120° to cyclic
Question 19.

Let us write whether the following statements are true or false:

(i) The angle, formed by rotating a ray centering its end point in anticlockwise direction is positive.

(ii) The angle, formed for completely rotating a ray twice by centering its end point is 720°
Answer:

(i) True,

Positive angles are made by rotating anti-clockwise.


(ii) True,


On one rotation angle is 360°


∴ 360° × 2 = 720°
Question 20.

Let us fill in the blanks:

(i) π radian is a ______angle.

(ii) In sexagesimal system 1 radian equals to ______(approx)

(iii) The circular value of the supplementary angle of the measure is_____
Answer:

(i) circular

Radian are denotation for circular angles.


(ii) 57.29



=57.29


(iii) 0


Sum of supplementary angles is π
Question 21.

If the value of an angle in degree is D and in radian is R; then let us determine the value of


Answer:




Question 22.

Let us write the value of complementary angle of the measure 63°35’15’’
Answer:

63° + 35’ + 15’’
Question 23.

If the measures of two angles of a triangle are 65°56’55’’ and 64°3’5’’, then let us determine the circular value of third angle.
Answer:

65°56’55’’



In radians



= 1.064


Angle 2


64°3’5’’




In radians



= 1.130


Third angle


x = π – 1.130 – 1.064


⇒ x = 0.9476
Question 24.

In a circle, if an arc of 220 cm. length subtends an angle of measure 63° at the centre, then let us determine the radius of the circle.
Answer:

converting 63° to radians


Taking




Let the radius be r.



⇒ r = 200 cm
Question 25.

Let us write the circular value of an angle formed by the end point of hour hand of a clock in 1 hour rotation.
Answer:

In one complete circle of 12 hours

It completes 2π angle


⇒ In 1 hour, it’ll complete