A triangular plot of land has sides of lengths 410 feet, 360 feet, and 170 feet. Approximate the smallest angle (in degrees) between the sides. (Round your answer to the nearest whole number.)
the smallest angle will be the one opposite the shortest side
if a = 410 and b = 360 and c = 170 we want angle C
but we all know the law of cosines
c^2 = a^2 + b^2 - 2 a b cos C
170^2 = 410^2 + 360^2 - 2*410*360 cos C
solve for cosC
then find cos^-1 C
the law of cosines gives this quickly:
ie: the angle between the 410 and 360 sides is
170^2=410^1+360^2 - 2*410*360*cos Angle1
solve for angle1
then get a second angle angle 2
360^2=410^1+170^2-2*410*170*cos2
solve for cosine 2
then knowing all three angles add to 180, solve for angle 3
After you have done that, make a sketch with just three sides. You wlll come to the conclusion that the smallest angle will always be between the two longest legs (angle1 here).
To approximate the smallest angle between the sides of a triangle, we can use the Law of Cosines. The Law of Cosines states that the square of one side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those sides multiplied by the cosine of the included angle.
In this case, let's call the sides of the triangle A, B, and C, with angles opposite A, B, and C respectively.
According to the problem, side A has a length of 410 feet, side B has a length of 360 feet, and side C has a length of 170 feet.
We want to find the smallest angle, which is the angle opposite side C.
Using the Law of Cosines, we can determine the cosine of the smallest angle:
cos(C) = (A² + B² - C²) / (2 * A * B)
Plugging in the given values, we have:
cos(C) = (410² + 360² - 170²) / (2 * 410 * 360)
Now we can use the inverse cosine function (cos⁻¹) to find the angle:
C = cos⁻¹ ((410² + 360² - 170²) / (2 * 410 * 360))
Calculating this expression, we find that C is approximately 34.7 degrees.
Rounding this to the nearest whole number, the smallest angle between the sides of the triangular plot of land is 35 degrees.