## To determine the initial angle at which the range is equal to the maximum height in the special case of projectile motion, we can set the formulas for range and maximum height equal to each other and solve for the angle (A).

As given, the maximum height (H) achieved is given by:

H = (1/2)(V^2/g)sin^2(A),

where V represents the launch velocity and g is the acceleration due to gravity.

The range (R) is given by:

R = 2(V^2/g)sin(A)cos(A).

To find the angle (A) at which the range equals the maximum height, we set R equal to H and solve for A:

2(V^2/g)sin(A)cos(A) = (1/2)(V^2/g)sin^2(A).

We can simplify this equation by dividing through by (V^2/g):

2sin(A)cos(A) = (1/2)sin^2(A).

Now, we can simplify further by replacing cos(A) with sqrt(1 - sin^2(A)) (using the identity cos^2(A) + sin^2(A) = 1):

2sin(A)sqrt(1 - sin^2(A)) = (1/2)sin^2(A).

Squaring both sides of the equation, we get:

4sin^2(A)(1 - sin^2(A)) = (1/4)sin^4(A).

Expanding and rearranging the terms, we have:

4sin^2(A) - 4sin^4(A) = (1/4)sin^4(A).

Multiplying through by 4, we obtain:

16sin^2(A) - 16sin^4(A) = sin^4(A).

Combining like terms, we have:

17sin^4(A) - 16sin^2(A) = 0.

Now, we can factor out sin^2(A) from both terms:

sin^2(A)(17sin^2(A) - 16) = 0.

To find the possible values for sin(A), we set each factor equal to zero:

sin^2(A) = 0,

and

17sin^2(A) - 16 = 0.

Solving sin^2(A) = 0 gives us sin(A) = 0, which gives us an angle of A = 0 degrees or A = 180 degrees.

For the equation 17sin^2(A) - 16 = 0, we can rearrange it as:

17sin^2(A) = 16.

Dividing through by 17, we have:

sin^2(A) = 16/17.

Taking the square root of both sides, we get:

sin(A) = sqrt(16/17),

which gives us two possible values for sin(A):

sin(A) = sqrt(16/17) or sin(A) = -sqrt(16/17).

Taking the inverse sine (or arcsine) of both sides, we find the corresponding angles:

A = arcsin(sqrt(16/17)) or A = -arcsin(sqrt(16/17)).

Now, we have found multiple values of A that satisfy the equation. To determine the specific initial angle at which the range equals the maximum height, we would need to know the constraints or given conditions of the problem. These may include restrictions on the range of angles or the direction of motion.

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