If b=3,c=4 and angle B=60.degrees,then the number of triangles that can be contacted is

A-infinite, B-two ,C-one ,D-nil

To determine how many triangles can be constructed given the values b = 3, c = 4, and angle B = 60 degrees, we need to apply the Law of Sines and some geometric principles. Let's break this down step by step.

Understanding Triangle Construction

When constructing a triangle with two sides and the included angle known, we can use the Law of Sines to find the other angles and the third side. In this case, we have:

• Side b = 3 (opposite angle B)
• Side c = 4 (opposite angle C)
• Angle B = 60 degrees

Applying the Law of Sines

The Law of Sines states that the ratio of a side to the sine of its opposite angle is constant for all three sides of a triangle. This can be written as:

sin(A)/a = sin(B)/b = sin(C)/c

In our case, we can set up the relationship:

sin(B)/b = sin(C)/c

sin(60°)/3 = sin(C)/4

Calculating sin(60°)

Since sin(60°) equals √3/2, we can rewrite the equation as follows:

(√3/2) / 3 = sin(C) / 4

Solving for sin(C)

From this, we can cross-multiply to find sin(C):

4(√3/2) = 3sin(C)

This simplifies to:

2√3 = 3sin(C)

Now, solving for sin(C) gives us:

sin(C) = (2√3)/3

Determining the Possible Values for Angle C

The sine function can produce two angles in the range of 0° to 180° for a given value, except when it reaches 0 or 1. Since sin(C) = (2√3)/3 is less than 1, we can have two possible angles for C:

• C₁ = sin⁻¹((2√3)/3) (acute angle)
• C₂ = 180° - C₁ (obtuse angle)

Angle A Calculation

Since the angles in a triangle must sum up to 180°, we can find angle A for both cases:

A₁ = 180° - B - C₁
A₂ = 180° - B - C₂

This means that we can indeed form two distinct triangles, each corresponding to one of the possible values for angle C.

Final Conclusion

Based on the calculations, the answer to your question is that the number of triangles that can be constructed under these conditions is two. Therefore, the correct option is B - two. This scenario highlights the fascinating aspect of triangles where specific side lengths and angles yield multiple valid configurations.