ABC is a triangle in vertical plane. Its two base angles BAC and BCA are 45º and tan–1 (1/3) respectively. A particle is projected from point A such that it passes through vertices B and C. Angle of projection is

To determine the angle of projection from point A such that the particle passes through points B and C in triangle ABC, we first need to analyze the triangle's geometry and the angles involved. Given that angle BAC is 45º and angle BCA is tan⁻¹(1/3), we can find the third angle ABC using the triangle angle sum property, which states that the sum of the angles in a triangle is always 180º.

Finding the Third Angle

Let's denote the angles as follows:

• ∠BAC = 45º
• ∠BCA = tan⁻¹(1/3)
• ∠ABC = ?

Using the triangle angle sum property:

∠ABC = 180º - ∠BAC - ∠BCA

First, we need to calculate ∠BCA:

Let ∠BCA = θ, where θ = tan⁻¹(1/3). This means that the tangent of angle θ is 1/3. We can find the sine and cosine of θ using the definition of tangent:

tan(θ) = opposite/adjacent = 1/3. This can be represented as a right triangle where the opposite side is 1 and the adjacent side is 3. Using the Pythagorean theorem, we can find the hypotenuse:

hypotenuse = √(1² + 3²) = √(1 + 9) = √10.

Thus, we have:

• sin(θ) = opposite/hypotenuse = 1/√10
• cos(θ) = adjacent/hypotenuse = 3/√10

Calculating ∠ABC

Now substituting the values into the angle sum equation:

∠ABC = 180º - 45º - tan⁻¹(1/3)

To find the angle of projection, we need to use the concept of projectile motion. The angle of projection θ₀ can be determined using the relationship between the angles and the distances involved.

Using Projectile Motion Principles

When a particle is projected from point A, it follows a parabolic trajectory. The angle of projection can be calculated using the formula:

tan(θ₀) = (h - h₀) / d

Where:

• h is the height of point B or C above point A.
• h₀ is the initial height (which is 0 if we consider point A as the origin).
• d is the horizontal distance from A to the line connecting B and C.

Applying the Angles

Since we have the angles, we can express the height and distance in terms of the angles:

Let’s assume the distance from A to B is d₁ and from A to C is d₂. The height at B can be expressed as:

h_B = d₁ * tan(∠BAC) = d₁ * tan(45º) = d₁.

For point C, the height can be expressed as:

h_C = d₂ * tan(∠BCA) = d₂ * (1/3).

Final Calculation of Angle of Projection

To find the angle of projection θ₀, we can use the relationship derived from the heights and distances:

tan(θ₀) = (h_C - h_B) / (d₂ - d₁).

Substituting the values we have:

tan(θ₀) = [(d₂ * (1/3)) - d₁] / (d₂ - d₁).

From this equation, we can solve for θ₀, which gives us the angle of projection needed for the particle to pass through points B and C. The exact numerical value will depend on the specific distances d₁ and d₂, but the method outlined here provides a clear pathway to find the angle of projection based on the triangle's geometry and the principles of projectile motion.