The vertical angle of a cone is 70 degrees. And it’s slant angle of the cone
is not provided, therefore, it cannot be determined. We need the radius or the height of the cone to determine the slant angle using trigonometry.
To find the slant angle of a cone, we need to use trigonometric ratios. The slant height (l) is related to the vertical height (h) and the radius (r) of the cone by the equation:
l^2 = h^2 + r^2
First, we need to find the vertical height (h) of the cone. Since we know the vertical angle (θ) is 70 degrees, we can use the following trigonometric ratio:
sin(θ) = h / l
Rearranging the equation, we have:
h = l * sin(θ)
Now, substituting this value of h into the equation for the slant height, we get:
l^2 = (l * sin(θ))^2 + r^2
Expanding the equation, we have:
l^2 = l^2 * sin^2(θ) + r^2
Simplifying the equation, we get:
l^2 - l^2 * sin^2(θ) = r^2
Factoring out l^2, we get:
l^2 * (1 - sin^2(θ)) = r^2
Using the trigonometric identity sin^2(θ) + cos^2(θ) = 1, we can rewrite the equation as:
l^2 * cos^2(θ) = r^2
Now, we can solve for the slant angle (α) using the following trigonometric ratio:
cos(α) = r / l
Taking the inverse cosine of both sides, we get:
α = cos^(-1)(r / l)
Thus, the slant angle (α) of the cone is equal to the inverse cosine of the ratio of the radius (r) to the slant height (l).