Question 1

Step 1: Identify the given information and choose the correct law.

Accepted Answer

We are given $a = 120.0$, $b = 85.0$, and $C = 42.0^\circ$. Because we do not have a matching side-angle pair (we have angle $C$ but not side $c$), we cannot use the Law of Sines. We must start with the Law of Cosines to find side $c$:
$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$

Question 2

Step 2: Calculate the missing side.

Accepted Answer

Plug in the known values:
$$c^2 = 120^2 + 85^2 - 2(120)(85) \cdot \cos(42^\circ)$$
$$c^2 = 14400 + 7225 - 20400 \cdot 0.7431448$$
$$c^2 = 21625 - 15160.15$$
$$c^2 = 6464.85$$
Take the square root of both sides:
$$c = \sqrt{6464.85} = 80.404 	ext{ meters}$$

Question 3

Step 3: Calculate the second angle.

Accepted Answer

Now we know side $c = 80.404$ and angle $C = 42^\circ$. We have a matching pair, so we can use the Law of Sines to find angle $B$. 
$$\frac{\sin(B)}{b} = \frac{\sin(C)}{c}$$
$$\frac{\sin(B)}{85} = \frac{\sin(42^\circ)}{80.404}$$
$$\sin(B) = \frac{85 \cdot 0.66913}{80.404} = \frac{56.876}{80.404} = 0.70738$$
To find the angle, take the inverse sine (arcsin) of $0.70738$:
$$B = \arcsin(0.70738) = 45.02^\circ$$

Question 4

Step 4: Calculate the final angle.

Accepted Answer

We now know angle $C = 42^\circ$ and angle $B = 45.02^\circ$. The sum of all angles is $180^\circ$.
$$A = 180^\circ - 42^\circ - 45.02^\circ = 92.98^\circ$$

Question 5

Can you solve a triangle if you only know the three angles (AAA)?

Accepted Answer

No, it is mathematically impossible to solve for the side lengths of a triangle using only three angles. Knowing the three angles defines the shape and proportions of the triangle, but it provides zero information about its scale. An equilateral triangle with angles of $60^\circ-60^\circ-60^\circ$ could have sides of 1 inch, 100 feet, or 50 miles. To determine the specific size of a triangle, you must be given at least one side length to anchor the measurements.

Question 6

What is the difference between plane trigonometry and spherical trigonometry?

Accepted Answer

Plane trigonometry deals with triangles drawn on a perfectly flat, two-dimensional surface, where the interior angles always sum to exactly $180^\circ$. Spherical trigonometry deals with triangles drawn on the surface of a sphere, like the Earth. On a sphere, the sides of a triangle are curved arcs (great circles), and the sum of the interior angles is always strictly greater than $180^\circ$ and less than $540^\circ$. Spherical trigonometry requires entirely different formulas and is used primarily in global navigation and astronomy.

Question 7

How do I know if the triangle measurements I was given are impossible?

Accepted Answer

There are two primary ways a set of given measurements can be physically impossible. First, they might violate the Triangle Inequality Theorem, which states that the sum of the two shorter sides must be strictly greater than the longest side. If you are given sides 2, 3, and 10, no triangle can be formed ($2+3$ is not greater than 10). Second, in an SSA configuration, if the given side opposite the given angle is shorter than the calculated altitude ($h = b \cdot \sin(A)$), the side cannot physically reach the base, making the triangle impossible.

Question 8

Why do we need both the Law of Sines and the Law of Cosines?

Accepted Answer

Each law has a specific mathematical limitation that the other law solves. The Law of Sines requires you to know at least one "matching pair" of a side and its opposite angle to establish the baseline ratio. If you are given three sides (SSS) or two sides and an included angle (SAS), you do not have a matching pair, rendering the Law of Sines useless. The Law of Cosines does not require a matching pair, allowing you to bridge the gap and find the first missing piece, after which you can use either law to finish the job.

Question 9

Does Heron's Formula work for right triangles?

Accepted Answer

Yes, Heron's Formula works perfectly for absolutely any type of triangle—right, acute, obtuse, scalene, or equilateral—as long as you know the exact lengths of all three sides. While it is faster to use $Area = 0.5 \cdot 	ext{base} \cdot 	ext{height}$ for a right triangle (since the two legs act as the base and height), Heron's formula will yield the exact same mathematical result. It is a universal geometric truth.

Question 10

What happens if I use the Law of Sines to find an obtuse angle?

Accepted Answer

If you use the Law of Sines to calculate an angle that is actually obtuse (greater than $90^\circ$), your calculator will give you the wrong answer. The inverse sine function ($\arcsin$) is mathematically restricted to returning values between $-90^\circ$ and $+90^\circ$. If the true angle is $120^\circ$, the calculator will output $60^\circ$ (because $\sin(120^\circ) = \sin(60^\circ)$). To avoid this dangerous trap, experts always use the Law of Cosines to find the largest angle in a triangle, because the inverse cosine function ($\arccos$) safely returns values up to $180^\circ$.

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