In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?
Used Formula:
sin 2θ = 2 sin θ cos θ
cos C + cos D = 2cos((C + D)/2) cos((C − D)/2)
Calculation:
sin A − cos B − cos C = 0
⇒ sin A = cos B + cos C
Using the formula given above
⇒ 2 sin(A/2) cos(A/2) = 2cos((B + C)/2) cos((B − C)/2)
We know that, for ΔABC
∠A + ∠B + ∠C = π
⇒ sin(A/2) cos(A/2) = cos((π − A)/2) cos((B − C)/2)
Since, cos(π/2 − θ) = sin θ
⇒ sin(A/2) cos(A/2) = sin(A/2) cos((B − C)/2)
⇒ cos(A/2) = cos((B − C)/2)
⇒ A/2 = (B − C)/2
⇒ A = B − C
⇒ B = A + C
But ∠A + ∠B + ∠C = π
⇒ B = π − B
⇒ 2B = π
⇒ B = π/2
∴ Angle B is equal to π/2.
Similar Questions
In a triangle ABC, a = (1 + √3) cm, b = 2 cm and angle C = 60°, then the other two angles are
Consider the following statements :
1. ABC is right angled triangle
2. The angles of the triangle are in AP
Which of the statements given above is/are correct ?
If c = 8, what is the area of the triangle ?
What is the value of a + b + √2 c equal to ?
What is the ratio of a2 ∶ b2 ∶ c2 ?
Consider the following statements:
1. If ABC is a right-angled triangle, right-angled at A, and if sin \(\rm B = \frac 1 3,\) then cosec C = 3.
2. If b cos B = c cos C and if the triangle ABC is not right-angled, then ABC must be isosceles.
Which of the above statements is/are correct?
If the angles of a triangle ABC are in AP and b : c = √3 : √2, then what is the measure of angle A?
In a triangle ABC if a = 2, b = 3 and sin A = 2/3, then what is angle B equal to?
The sides of a triangle are m, n and \(\rm \sqrt{m^2+n^2+mn}\) . What is the sum of the acute angles of the triangle?
In a triangle ABC, a – 2b + c = 0. The value of \(\cot \left( {\frac{A}{2}} \right)\cot \left( {\frac{C}{2}} \right)\) is
Important Questions from Properties of Triangles
In a triangle ABC, a = (1 + √3) cm, b = 2 cm and angle C = 60°, then the other two angles are
Which of the following measures can form a triangle?
Which of the following cannot be the sides of a triangle?
If in a triangle ABC, \(\frac{{2\cos A}}{a} + \frac{{\cos B}}{b} + \frac{{2\cos C}}{c} = \frac{a}{{bc}} + \frac{b}{{ca}}\) then the value of the angle A is
If the data given to construct a triangle ABC are a = 5, b = 7, \(\sin A = \frac{3}{4}\), then it is possible to construct
