# A triangle has two corners with angles of # (3 pi ) / 4 # and # ( pi )/ 6 #. If one side of the triangle has a length of #3 #, what is the largest possible area of the triangle?

Largest possible area of the triangle is 6.1471

The remaining angle:

I am assuming that length AB (3) is opposite the smallest angle.

Using the ASA

By signing up, you agree to our Terms of Service and Privacy Policy

To find the largest possible area of the triangle, we'll use the formula for the area of a triangle given two sides and the angle between them, which is ( A = \frac{1}{2}ab\sin(C) ), where (a) and (b) are the lengths of the two sides and (C) is the angle between them.

Given that one side of the triangle has a length of 3, we need to find the lengths of the other two sides using trigonometry.

Using the Law of Sines, we can relate the lengths of the sides to the angles:

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

Let's denote the lengths of the sides as (a), (b), and (c) respectively.

We're given that one angle is ( \frac{3\pi}{4} ) and the other is ( \frac{\pi}{6} ). Let's find the third angle first:

[ \text{Third angle} = \pi - \left(\frac{3\pi}{4} + \frac{\pi}{6}\right) ]

[ = \pi - \frac{9\pi}{12} - \frac{2\pi}{12} ]

[ = \pi - \frac{11\pi}{12} ]

Now, using the Law of Sines, we can find the lengths of the other two sides:

[ \frac{a}{\sin\left(\frac{3\pi}{4}\right)} = \frac{3}{\sin\left(\frac{\pi}{6}\right)} ]

Solve for (a):

[ a = 3\sin\left(\frac{3\pi}{4}\right) ]

Similarly, we can find (b) using the same method:

[ b = 3\sin\left(\frac{\pi}{6}\right) ]

Now, we can find the area of the triangle using the formula ( A = \frac{1}{2}ab\sin(C) ), where (C) is the third angle we found earlier.

[ A = \frac{1}{2}(3\sin\left(\frac{3\pi}{4}\right))(3\sin\left(\frac{\pi}{6}\right))\sin\left(\pi - \frac{11\pi}{12}\right) ]

[ A = \frac{1}{2}(3\sin\left(\frac{3\pi}{4}\right))(3\sin\left(\frac{\pi}{6}\right))\sin\left(\frac{\pi}{12}\right) ]

Now, we can calculate the value of (A). Once we have (A), we can determine its maximum value.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Two corners of an isosceles triangle are at #(4 ,3 )# and #(9 ,3 )#. If the triangle's area is #64 #, what are the lengths of the triangle's sides?
- How do you use Heron's formula to find the area of a triangle with sides of lengths #4 #, #6 #, and #5 #?
- An ellipsoid has radii with lengths of #14 #, #7 #, and #6 #. A portion the size of a hemisphere with a radius of #9 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?
- How does the circumference of a circle change when its diameter is tripled?
- Two corners of an isosceles triangle are at #(5 ,2 )# and #(2 ,1 )#. If the triangle's area is #7 #, what are the lengths of the triangle's sides?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7