Given a triangle with side lengths: root3, root3, 3, find the measure of its smallest angle.
well,
tan(x/2) = 2/√3
now just take 2*arctan(2/√3)
Since it is isosceles, we can construct a right-angled triangle with an altitude to the base of 3
then cosØ = 1.5/√3 = 3/2√3
= 3/2√3 * √3/√3
= √3 /2
ahhh, do you recognize the 30-60-90° triangle
(if not, use your calculator)
Ø = 30°
or
Dang. I used x as the vertex angle. Go with Reiny.
To find the measure of the smallest angle in a triangle, we can use the Law of Cosines. The Law of Cosines states that for a triangle with side lengths a, b, and c, the square of one side (c) is equal to the sum of the squares of the other two sides (a and b) minus twice the product of the lengths of those two sides multiplied by the cosine of the included angle (θ).
In this case, we have a triangle with side lengths √3, √3, and 3. Let's call the side opposite the smallest angle a, the side opposite the second smallest angle b, and the side opposite the largest angle c. We want to find the smallest angle, so we can label it θ.
Using the Law of Cosines, we can write the equation as follows:
3^2 = (√3)^2 + (√3)^2 - 2(√3)(√3)cosθ
Simplifying the equation:
9 = 3 + 3 - 6cosθ
9 = 6 - 6cosθ
6cosθ = 6 - 9
6cosθ = -3
cosθ = -3/6
cosθ = -1/2
To find the value of θ, we can use the inverse cosine function (cos^(-1)):
θ = cos^(-1)(-1/2)
Using a calculator or a trigonometric table, we find that cos^(-1)(-1/2) is 120 degrees.
Therefore, the measure of the smallest angle in the given triangle is 120 degrees.