In triangle RST, RS=4, RT=7, and ST=5. Determine the measures of the angles to the nearest degree.

I solved correctly for T's angle of 34 degrees. Once I try using the sine law to solve for angle S, I keep getting 78 degrees, when it's supposed to be 102 degrees. I tried solving for R first, and it worked that way using sine law, but why won't solving for angle S first work out?

You are running into what is called the "ambiguous case" . For example

sin 30° = sin 150° = .5
because the sine is positive in the II and III quadrant.

So If you start with sinØ = .5
your calculator is programmed to always give the closest angle to 0, and that would be 30°

You probably used the cosine law to find the first angle. A suggestion is to always find the largest angle first. If that happens to be an obtuse angle, your calculator will find the correct angle.
Of course there can only be one obtuse angle in a triangle, so using the sine law for your second angle does not result in any problems

notice that in your solution 78 and 102 add up to 180 , and sin 78 = sin(180-78) = sin 102

I would have found angle S first, the largest angle always opposite the largest side.

7^2 = 5^2 + 4^2 - 2(5)(4)cosØ
40cosØ = -8
cosØ = -8/40 = -1/5
Ø = 101.537°

Now you can use the sine law and you will find the correct other angles.

To determine the measures of the angles in triangle RST, we can use the Law of Sines. Let's see why solving for angle S first does not work out.

Using the Law of Sines, we have:

sin(R) / RS = sin(S) / ST
sin(R) / 4 = sin(S) / 5

Now, let's solve for angle R first:

sin(R) = (4/7) * sin(T)

We can substitute the known values we have:

sin(R) = (4/7) * sin(34°)

By solving for R, we find R ≈ 46°.

Now, let's solve for angle S using the sine law:

sin(S) = (5/7) * sin(T)

Substituting the known values we have:

sin(S) = (5/7) * sin(34°)

By solving for S, we find S ≈ 102°.

Therefore, the measures of the angles in triangle RST (to the nearest degree) are:
Angle R ≈ 46°
Angle S ≈ 102°
Angle T = 34°

If you obtained a different result for angle S using the same methodology, it might be due to a calculation error. Please double-check your calculations to ensure accuracy.

To solve for the angles in triangle RST using the sine law, you need to make sure you are using the correct side ratios. The sine law states that the ratio of the length of a side of a triangle to the sine of its opposite angle is always the same.

Let's look at your case:

RS = 4, RT = 7, and ST = 5

You correctly solved for angle T using the sine law. Now, let's try solving for angle S.

To find angle S, we can use the sine law by setting up the following ratio:

Sin(S) / 5 = Sin(34) / 7

To solve for Sin(S), we can cross-multiply:

Sin(S) = 5 * Sin(34) / 7

Calculating this expression gives: Sin(S) ≈ 0.435

Now, to find the measure of angle S, you can take the inverse sine (also known as arcsine) of this value:

S ≈ arcsin(0.435) ≈ 26.1 degrees

It seems you obtained angle S as 78 degrees, which is not correct. This suggests that there may have been an error in your calculations.

On the other hand, if you solve for angle R first using the sine law, you would set up the following ratio:

Sin(R) / 4 = Sin(34) / 7

Again, cross-multiplying and solving for Sin(R) gives:

Sin(R) = 4 * Sin(34) / 7

Calculating this expression gives: Sin(R) ≈ 0.290

Taking the inverse sine of this value gives you:

R ≈ arcsin(0.290) ≈ 17.1 degrees

Finally, to find angle S, you can use the fact that the sum of the angles in a triangle is 180 degrees:

S = 180 - T - R = 180 - 34 - 17.1 ≈ 128.9 degrees, which can be rounded to 129 degrees.

Therefore, the correct measures of the angles are approximately:

T ≈ 34 degrees
R ≈ 17.1 degrees
S ≈ 129 degrees

I hope this clarifies why solving for angle S first did not work in your case. In general, it is recommended to solve for angles one by one and use known angles to find the remaining ones.