RD Sharma Class 11 Solutions Chapter - Sine and Cosine Formulae and their Applications | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


10. Sine and Cosine Formulae and their Applications
Exercise 10.1
1. Question
If in a ?ABC, ?A = 45
o
, ?B = 60
o
, and ?C = 75
o
; find the ratio of its sides.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
(? sin (a + b) = sin a cos b + sin b cos a)
Now substituting the corresponding values, we get,
Multiplying 2v2, we get
Hence the ratio of the sides of the given triangle is 
2. Question
If in any ?ABC, ?C = 105
o
, ?B = 45
o
, a = 2, then find b.
Answer
Page 2


10. Sine and Cosine Formulae and their Applications
Exercise 10.1
1. Question
If in a ?ABC, ?A = 45
o
, ?B = 60
o
, and ?C = 75
o
; find the ratio of its sides.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
(? sin (a + b) = sin a cos b + sin b cos a)
Now substituting the corresponding values, we get,
Multiplying 2v2, we get
Hence the ratio of the sides of the given triangle is 
2. Question
If in any ?ABC, ?C = 105
o
, ?B = 45
o
, a = 2, then find b.
Answer
We know in a triangle,
?A + ?B + ?C = 180°
? ?A = 180° - ?B - ?C
Substituting the given values, we get
?A = 180° - 45° - 105°
? ?A = 30°
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the corresponding values we get
Substitute the equivalent values of the sine, we get
Hence the value of b is 2v2 units.
3. Question
In ?ABC, if a = 18, b = 24 and c = 30 and ?C = 90
o
, find sin A, sin B and sin C.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Page 3


10. Sine and Cosine Formulae and their Applications
Exercise 10.1
1. Question
If in a ?ABC, ?A = 45
o
, ?B = 60
o
, and ?C = 75
o
; find the ratio of its sides.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
(? sin (a + b) = sin a cos b + sin b cos a)
Now substituting the corresponding values, we get,
Multiplying 2v2, we get
Hence the ratio of the sides of the given triangle is 
2. Question
If in any ?ABC, ?C = 105
o
, ?B = 45
o
, a = 2, then find b.
Answer
We know in a triangle,
?A + ?B + ?C = 180°
? ?A = 180° - ?B - ?C
Substituting the given values, we get
?A = 180° - 45° - 105°
? ?A = 30°
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the corresponding values we get
Substitute the equivalent values of the sine, we get
Hence the value of b is 2v2 units.
3. Question
In ?ABC, if a = 18, b = 24 and c = 30 and ?C = 90
o
, find sin A, sin B and sin C.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
Similarly,
Now substituting the given values we get
And given ?C = 90°, so sin C = sin 90° = 1.
Hence the values of sin A, sin B, sin C are  respectively
4. Question
In any triangle ABC, prove the following:
Answer
Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get
?a = k sin A
Similarly, b = k sin B
So, a - b = k(sin A - sin B)
And a + b = k(sin A + sin B)
Page 4


10. Sine and Cosine Formulae and their Applications
Exercise 10.1
1. Question
If in a ?ABC, ?A = 45
o
, ?B = 60
o
, and ?C = 75
o
; find the ratio of its sides.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
(? sin (a + b) = sin a cos b + sin b cos a)
Now substituting the corresponding values, we get,
Multiplying 2v2, we get
Hence the ratio of the sides of the given triangle is 
2. Question
If in any ?ABC, ?C = 105
o
, ?B = 45
o
, a = 2, then find b.
Answer
We know in a triangle,
?A + ?B + ?C = 180°
? ?A = 180° - ?B - ?C
Substituting the given values, we get
?A = 180° - 45° - 105°
? ?A = 30°
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the corresponding values we get
Substitute the equivalent values of the sine, we get
Hence the value of b is 2v2 units.
3. Question
In ?ABC, if a = 18, b = 24 and c = 30 and ?C = 90
o
, find sin A, sin B and sin C.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
Similarly,
Now substituting the given values we get
And given ?C = 90°, so sin C = sin 90° = 1.
Hence the values of sin A, sin B, sin C are  respectively
4. Question
In any triangle ABC, prove the following:
Answer
Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get
?a = k sin A
Similarly, b = k sin B
So, a - b = k(sin A - sin B)
And a + b = k(sin A + sin B)
So, the given LHS becomes,
But,
Substituting the above values in equation (i), we get
Rearranging the above equation we get,
Hence proved
5. Question
In any triangle ABC, prove the following:
Answer
Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get
?a = k sin A
Similarly, b = k sin B
So, a - b = k(sin A - sin B)..(i)
So the given LHS becomes,
Page 5


10. Sine and Cosine Formulae and their Applications
Exercise 10.1
1. Question
If in a ?ABC, ?A = 45
o
, ?B = 60
o
, and ?C = 75
o
; find the ratio of its sides.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
(? sin (a + b) = sin a cos b + sin b cos a)
Now substituting the corresponding values, we get,
Multiplying 2v2, we get
Hence the ratio of the sides of the given triangle is 
2. Question
If in any ?ABC, ?C = 105
o
, ?B = 45
o
, a = 2, then find b.
Answer
We know in a triangle,
?A + ?B + ?C = 180°
? ?A = 180° - ?B - ?C
Substituting the given values, we get
?A = 180° - 45° - 105°
? ?A = 30°
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the corresponding values we get
Substitute the equivalent values of the sine, we get
Hence the value of b is 2v2 units.
3. Question
In ?ABC, if a = 18, b = 24 and c = 30 and ?C = 90
o
, find sin A, sin B and sin C.
Answer
Let a, b, c be the sides of the given triangle. Then by applying the sine rule, we get
Now substituting the given values we get
Similarly,
Now substituting the given values we get
And given ?C = 90°, so sin C = sin 90° = 1.
Hence the values of sin A, sin B, sin C are  respectively
4. Question
In any triangle ABC, prove the following:
Answer
Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get
?a = k sin A
Similarly, b = k sin B
So, a - b = k(sin A - sin B)
And a + b = k(sin A + sin B)
So, the given LHS becomes,
But,
Substituting the above values in equation (i), we get
Rearranging the above equation we get,
Hence proved
5. Question
In any triangle ABC, prove the following:
Answer
Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get
?a = k sin A
Similarly, b = k sin B
So, a - b = k(sin A - sin B)..(i)
So the given LHS becomes,
Substituting equation (i) in above equation, we get
But,
Substituting the above values in equation (ii), we get
Rearranging the above equation we get
But 
So the above equation becomes,
But from sine rule,
So the above equation becomes,
Hence proved
6. Question
In any triangle ABC, prove the following:
Answer