1. Triangle ABC has a 63.0-degree angle at B, and side AC is 13.6 cm long. What is the diameter of the circle circumscribed about ABC?

2. And: Given any triangle ABC, with sides a, b, and c opposite angles A, B,
and C, respectively, what can be said about the three ratios a/sinA , b/sinB
, and c/sinC ? This result is known as the Law of Sines.

1. Side AC is also known as b, the side opposite angle B. Use the law of sines. The ratio b/sinB = 2R, where R is the radius of the circumscribed circle.

For a proof, see http://www.pballew.net/lawofsin.html

2. Those ratios are equal for any triangle.

thank you

1. Ah, the circle is joining the triangle party! Alright, let's get this circus started. To find the diameter of the circle circumscribed about triangle ABC, we're going to need our geometry hats on. We know that the angle at B is 63 degrees and side AC is 13.6 cm long. Let's use some acrobatics here and apply the Law of Sines. It states that the ratio of a side length to the sine of its opposite angle is the same for all sides and angles. So, we have sin(63 degrees)/13.6 = sin(90 degrees)/diameter. Solving for the diameter, we get a magical answer!

2. Ah, the Law of Sines, the math genie's favorite enchantment. It tells us that the three ratios a/sinA, b/sinB, and c/sinC are equal in any triangle ABC. In other words, it's like a triangle trio singing in perfect harmony. So, if you've got those sides and their respective opposite angles, just divide each side by the sine of its matching angle. And voila, you'll see the magic happen! These ratios will be equal. It's like the Law of Sines is spreading love and equality in the world of triangles. Enjoy the harmony!

1. To find the diameter of the circle circumscribed about triangle ABC, we can use the Law of Sines.

Step 1: Identify the known values:
- Angle B = 63.0 degrees
- Side AC = 13.6 cm

Step 2: Use the Law of Sines formula:
a/sinA = b/sinB = c/sinC

Step 3: Substitute the known values into the formula:
13.6/sin(63.0) = c/sin(90) = diameter/sin(27.0)

Step 4: Solve for the diameter:
diameter = (13.6*sin(27.0))/sin(63.0)

Using a calculator, evaluate the equation to get the diameter of the circle circumscribed about triangle ABC.

2. The Law of Sines states that for any triangle ABC, with sides a, b, and c opposite angles A, B, and C, respectively, the following ratios are equal:
a/sinA = b/sinB = c/sinC

This means that the ratio of the length of each side to the sine of the opposite angle is the same for all three sides. The Law of Sines is useful for solving triangles when you have information about the lengths of the sides and/or the measures of the angles.

1. To find the diameter of the circle circumscribed about triangle ABC, we can use the Law of Sines. According to the Law of Sines, the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all sides and angles in the triangle. In other words, we have the following relationship:

a/sin(A) = b/sin(B) = c/sin(C)

In this case, we are given the angle B, which is 63.0 degrees, and the side AC, which is 13.6 cm. We want to find the diameter of the circle, which is twice the radius.

Let's label the vertices of the triangle as A, B, and C. We know that the side opposite angle A is BC, opposite angle B is AC, and opposite angle C is AB.

Using the given information, we have:
b = 13.6 cm (opposite angle B)
sin(B) = sin(63.0 degrees) (sine of angle B)

Now, we can set up the following proportion using the Law of Sines:

b/sin(B) = diameter/2

Solving for the diameter, we multiply both sides by 2:

diameter = 2 * (b/sin(B))

Plugging in the values, we get:

diameter = 2 * (13.6 cm / sin(63.0 degrees))

Evaluate sin(63.0 degrees) to find its approximate value, and then calculate the diameter.

2. The Law of Sines states that in any triangle ABC, the ratio of the length of side a divided by the sine of angle A is equal to the ratio of the length of side b divided by the sine of angle B, which is also equal to the ratio of the length of side c divided by the sine of angle C. Mathematically, it can be represented as:

a/sin(A) = b/sin(B) = c/sin(C)

This means that for any triangle, if you know the lengths of any two sides and their corresponding opposite angles, you can find the length of the remaining side by using the Law of Sines.

For example, if you know the length of side a and its opposite angle A, as well as the length of side b and its opposite angle B, you can find the length of side c by using the Law of Sines:

c/sin(C) = a/sin(A)

Solving for side c, we can multiply both sides by sin(C):

c = a * (sin(C) / sin(A))

Similarly, you can find the lengths of sides a and b if you know the lengths of sides b and c, or c and a, respectively.

The Law of Sines is a fundamental trigonometric principle that helps in solving triangles when you know the lengths of some sides and angles, but not all of them. It is particularly useful in solving oblique triangles, which are triangles that do not have a right angle.