Which Triangle has 2 solutions?

a. A=130 degrees, a=19, b=11
b. A=45 degrees, a= 4(sqrt 2) b=8
c. A=32 degrees, a=16, b= 21
d. A=90 degrees, a=25, b=15

I really don't know how to do this one so I can't put what my answer is.

You'll have to use the Ambiguous Case for the Law of Sines.

I would give you a few links to some good websites. However, I'm not allowed. Try googling "ambiguous case law sines" without the quotes for some relevant information.

To determine which triangle has 2 solutions, we need to use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant for all sides and angles in the triangle. Mathematically, it can be written as:

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

In this case, we are given the angle A and the lengths of sides a and b. To check if a triangle has 2 solutions, we need to determine whether the value of sin(B) is positive.

Let's go through the options one by one:

a. A=130 degrees, a=19, b=11
Here, we are given angle A as 130 degrees, side a as 19 and side b as 11. To find angle B, we can use the Law of Sines equation:

19/sin(130) = 11/sin(B)

Solving this equation, we find that sin(B) is negative, meaning there is no valid solution. Therefore, this option does not have 2 solutions.

b. A=45 degrees, a= 4(sqrt 2), b=8
Using the Law of Sines, we can write:

4(sqrt 2)/sin(45) = 8/sin(B)

Simplifying, we find that sin(B) = 4(sqrt 2)/8 = 1/√2. Since sin(B) is positive, this option does have 2 valid solutions.

c. A=32 degrees, a=16, b= 21
Applying the Law of Sines, we get:

16/sin(32) = 21/sin(B)

Solving for sin(B), we find that sin(B) is positive, so this option also has 2 valid solutions.

d. A=90 degrees, a=25, b=15
Using the Law of Sines equation, we have:

25/sin(90) = 15/sin(B)

Since the sine of 90 degrees is always 1, this equation simplifies to 25 = 15/sin(B), which has no valid solution. Therefore, this option does not have 2 solutions.

In conclusion, options b and c have 2 solutions.