Question

if in an acute angled triangle abc if sin A+B-C = 1/2 and cos B+C- A = 1/root2 find angles a b c

Answer 1

Angles of ‘A’, ‘B’ and ‘C’ are  $\bold{67.5^{\circ}, 37.5^{\circ}, 75^{\circ}}$

GIVEN:

A+B+C = 1/2  

To find:

Angles of A,B nd C

Solution:

Let us take the triangle ABC, now as the question says $A+B-C=\frac{1}{2}\ \text{and}\ \text{cos} B+C-A=\frac{1}{\sqrt{2}}$

Therefore, the values can be written, as

$\sin A+B-C=\frac{1}{2} ; \quad A+B-C=\sin ^{-1}\left(\frac{1}{2}\right) ; A+B-C=30$and for the cosine the value is $B+C-A=\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right) ; B+C-A=45$

So, as we know that the sum of all ‘sides of a triangle’ is 180,

Hence, $A + B + C = 180.$

$B+C-A=45;  A+B-C=30;A+B+C=180$

With all the equation, we can find the values of A, B and C. Equating all equation:

Let us take $A+B-C=30;A+B+C=180$

Substituting value of A+B=C+30 in A+B+C=180, we get $C+30+C=180;C=75^{\circ}$

Now with $C=75^{\circ}$ we can find the value of A and B, putting the value of C in A+B=C+30 we get $A+B=75+30; A+B=105$

And putting the value of $C\ \text{in}\ B+C-A=45$ we get B+75-A=45;  B-A=-30 or A-B=30

Equate A-B=30 & A+B=105 we get the value of A as $67.5^{\circ}$

Therefore, value of B is $37.5^{\circ}$

Hence, the angles of ‘A’, ‘B’ and ‘C’ are $67.5^{\circ}, 37.5^{\circ}, 75^{\circ}$

Answer 2

A = 67.5°

B = 37.5°

C = 75°

Given:

$\sin (A+B-C)=\frac{1}{2}$

$\cos (B+C-A)=\frac{1}{\sqrt{2}}$

Step-by-step explanation:

We know that in a triangle, sum of the angles = 180°

A+B+C = 180 → (1)

We know that,

$\sin 30=\frac{1}{2}$

$\cos 45=\frac{1}{\sqrt{2}}$

So,

sin (A+B-C) = sin 30

A+B-C = 30 → (2)

And

cos (B+C-A) = cos 45

B+C-A = 45 → (3)

On solving equation (1) and (2), we get,  

A+B+C-A-B+C = 180-30 = 150

2C = 150

C = 75°

Substituting C=75 in equation (2), we get,

A+B-75 = 30

A+B = 105 → (4)

Also, substituting in equation (3), we get,

B+75-A =45

A-B = 30 → (5)

Adding equations (4) and (5), we get,

2A = 135 → A = 67.5°

B = A-30 = 67.5 - 30 = 37.5°

Therefore, A=67.5°; B=37.5°; and C=75°