Determine Whether the Following Measurements Produce a Triangle :
Here we are going to see some example problems to determine whether the following measurements produce a triangle.
Question 1 :
Determine whether the following measurements produce one triangle, two triangles or no triangle:
∠B = 88°, a = 23, b = 2. Solve if solution exists.
Solution :
Let us apply the given details in sine formula
a/sin A = b/sin B = c/sin C
23/sin A = 2/sin 88° = c/sin C
23/sin A = 2/sin 88°
sin A = (23/2) sin 88°
sin A = 11.5 sin 88°
The maximum value of sin θ will be 1. So, it is not possible.
Hence the given measurements will not produce a triangle.
Question 2 :
If the sides of a triangle ABC are a = 4, b = 6 and c = 8, then show that 4 cosB + 3cosC = 2.
Solution :
Let us use cosine formula to find the values of cos B and cos C.
Cosine formula :
cos A = (b2 + c2 - a2)/2bc --------(1)
cos B = (a2 + c2 - b2)/2ac --------(2)
cos C = (a2 + b2 - c2)/2ab --------(3)
From (2)
cos B = (a2 + c2 - b2)/2ac
cos B = (42 + 82 - 62)/2(4)(8)
cos B = (16 + 64 - 36)/64
cos B = 44/64 = 11/16
From (3)
cos C = (a2 + b2 - c2)/2ab
cos C = (42 + 62 - 82)/2(4)(6)
= (16 + 36 - 64)/48
= -12/48
cos C = - 1/4
Given that :
4 cos B + 3cos C = 2
L.H.S :
= 4(11/16) + 3(-1/4)
= (11/4) - (3/4)
= (11 - 3)/4 = 8/4
= 2 --> R.H.S
Hence it is proved.
Question 3 :
In a triangle ABC, if a = √3 − 1, b = √3 + 1 and C = 60°, find the other side and other two angles
Solution :
cos C = (a2 + b2 - c2)/2ab
a = √3 − 1, b = √3 + 1
a2 = (√3 − 1)2 = 3 + 1 - 2√3 = 4 - 2√3
b = √3 + 1
b2 = (√3 + 1)2 = 3 + 1 + 2√3 = 4 + 2√3
By applying those values in cosine formula
cos 60 = (4 - 2√3 + 4 + 2√3 - c2)/2(√3 − 1)(√3 + 1)
(1/2) = (8 - c2)/2(2)
2 = 8 - c2
c2 = 8 - 2 = 6
c = √6
Sine formula :
a/sin A = b/sin B = c/sin C
(√3 − 1)/sin A = (√3 + 1)/sin B = √6/sin 60°
(√3 + 1)/sin B = √6/(√3/2)
(√3 + 1)/sin B = 2 √6/√3
(√3 + 1)/sin B = 6√2/3
(√3 + 1)/sin B = 2√2
sin B = (√3 + 1)/2√2
sin B = (√3/2) (1/√2) + (1/2)(1/√2)
= sin 60 cos 45 + cos 60 sin 45
= sin (60 + 45)
= sin 105
Hence angle B is 105.
A + B + C = 180
A + 105 + 60 = 180
A + 165 = 180
A = 180-165
A = 15
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