If in a triangle ABC, angle B =π/3 and sinA SinC =y . Find the set of all possible values of y.

To solve the problem involving triangle ABC with angle B equal to π/3 and the relationship sinA SinC = y, we need to delve into some trigonometric properties and the constraints of a triangle. Let's break this down step by step.

Understanding the Triangle's Angles

In any triangle, the sum of the interior angles is always π (or 180 degrees). Given that angle B = π/3, we can express the other angles as follows:

• Angle A = α
• Angle C = γ

Thus, we have:

α + γ + π/3 = π

This simplifies to:

α + γ = π - π/3 = 2π/3

Using the Sine Rule

From the sine rule, we know that:

sinA / a = sinB / b = sinC / c

Where a, b, and c are the sides opposite angles A, B, and C, respectively. We can express sinA and sinC in terms of sinB:

sinB = sin(π/3) = √3/2

Thus, we can write:

sinA = k * a and sinC = k * c, where k is a proportionality constant.

Finding the Relationship Between sinA and sinC

Using the angle sum property, we can express sinC in terms of sinA:

sinC = sin(2π/3 - A) = sin(2π/3)cosA - cos(2π/3)sinA

Knowing that sin(2π/3) = √3/2 and cos(2π/3) = -1/2, we can substitute these values:

sinC = (√3/2)cosA + (1/2)sinA

Expressing y in Terms of sinA

Now, substituting sinA and sinC into the equation y = sinA * sinC gives:

y = sinA * [(√3/2)cosA + (1/2)sinA]

Expanding this, we have:

y = (√3/2)sinA * cosA + (1/2)sin²A

Maximizing y

To find the maximum and minimum values of y, we can analyze the expression:

y = (√3/4)sin(2A) + (1/2)sin²A

Since sinA can vary from 0 to 1 (as A ranges from 0 to π/3), we can find the maximum value of y by substituting the extreme values of sinA.

Calculating the Range of y

When sinA = 0, y = 0. When sinA = 1 (which is not possible in this triangle since A cannot be π/2), we need to check values closer to π/3. The maximum occurs when sinA is at its peak within the triangle's constraints.

Using calculus or numerical methods, we can find the critical points of y and evaluate them to determine the maximum value. However, through analysis, we find that:

• Minimum value of y = 0
• Maximum value of y = 1/2 (when sinA = √3/2)

Final Set of Values

Thus, the set of all possible values of y, given the constraints of triangle ABC, is:

y ∈ [0, 1/2]

This means that y can take any value from 0 to 1/2, inclusive, depending on the specific angles A and C within the triangle.