Math  /  Trigonometry

QuestionSolve each triangle ABCA B C that exists. A=37.8a=3.5c=15.1A=37.8^{\circ} \quad a=3.5 \quad c=15.1
Select the correct choice below and, if necessary, fill in the answer boxes within the choice. A. There is only one possible solution for the triangle. The measurements for the remaining angles BB and CC and side bb are as follows. B=B= \square C=C= \square { }^{\circ} b=b= \square (Round to the nearest tenth as needed.) B. There are two possible solutions for the triangle. The measurements for the solution with the longer side bb are as follows. B1=C1=\mathrm{B}_{1}=\square^{\circ} \quad \mathrm{C}_{1}= \square b1=b_{1}= \square The measurements for the solution with the shorter side bb are as follows. B2=\mathrm{B}_{2}= \square C2=\mathrm{C}_{2}= \square b2=\mathrm{b}_{2}= \square (Round to the nearest tenth as needed.) C. There are no possible solutions for the triangle.

Studdy Solution

STEP 1

What is this asking? Given one angle (AA) and two sides (aa and cc) of a triangle, we need to find the other angle and side measurements, keeping in mind there might be two, one, or no solutions! Watch out! Remember the ambiguous case of the Law of Sines!
When we're given two sides and an angle not between them, sometimes we get two possible triangles.

STEP 2

1. Apply the Law of Sines
2. Find angle CC
3. Check for multiple solutions
4. Find angle BB
5. Find side bb

STEP 3

We **know** angle AA and side aa, and side cc, so we can use the Law of Sines to find angle CC.
The Law of Sines says sinAa=sinCc\frac{\sin A}{a} = \frac{\sin C}{c}.
Let's **plug in** what we know: sin37.83.5=sinC15.1\frac{\sin 37.8^{\circ}}{3.5} = \frac{\sin C}{15.1}.

STEP 4

To **get** sinC\sin C by itself, we can **multiply** both sides of the equation by 15.115.1: 15.1sin37.83.5=sinC15.1 \cdot \frac{\sin 37.8^{\circ}}{3.5} = \sin C.

STEP 5

Now, let's **crunch those numbers**: sinC15.10.61293.52.646\sin C \approx 15.1 \cdot \frac{0.6129}{3.5} \approx 2.646.

STEP 6

Uh oh! sinC\sin C is greater than 11!
But the sine of *any* angle can never be greater than 11.
This means there's **no triangle** that can be formed with these measurements.

STEP 7

There are **no possible solutions** for the triangle.
We choose option C.

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