Clear, step‑by‑step examples of solving problems using the Law of Sines and Cosines

Starting with simple triangle examples using the Law of Sines

Let’s start with some friendly, concrete examples of solving problems using the Law of Sines. The Law of Sines says that in any triangle \(ABC\):

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

where \(a, b, c\) are the side lengths opposite angles \(A, B, C\) respectively.

Think of it as a matching rule: each side always “goes with” its opposite angle through the sine function. That pairing is what powers many of the best examples of non‑right‑triangle problem solving.

Example of using the Law of Sines: Finding a missing side

Imagine a triangle where:

• Angle \(A = 40^\circ\)
• Angle \(B = 75^\circ\)
• Side \(a = 10\) ft (opposite angle \(A\))

You’re asked to find side \(b\), opposite angle \(B\).

Step 1: Check that you have an angle–angle–side (AAS) situation.
You know two angles and a non‑included side. That’s a classic situation for the Law of Sines.

Step 2: Write the Law of Sines using the pieces you know.

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Plug in the values:

\[
\frac{10}{\sin 40^\circ} = \frac{b}{\sin 75^\circ}
\]

Step 3: Solve for \(b\).

\[
b = 10 \cdot \frac{\sin 75^\circ}{\sin 40^\circ}
\]

Using a calculator (in degree mode), you get approximately:

\[
b \approx 10 \cdot \frac{0.9659}{0.6428} \approx 15.0 \text{ ft (rounded)}
\]

This is one of the cleanest examples of solving problems using the Law of Sines: you match side to angle, set up a ratio, and solve.

Real‑world style example of the Law of Sines: Measuring a river width

Picture this: you’re standing on one bank of a river and want to estimate how wide it is without crossing. You pick a point directly across the river (call it point C). Then you walk 150 ft along your bank to point B, forming triangle \(ABC\) with \(AC\) as the unknown river width.

You measure:

• Angle at \(A = 52^\circ\)
• Angle at \(B = 68^\circ\)
• Side \(AB = 150\) ft

Step 1: Find the third angle.
Angles in a triangle add to \(180^\circ\):

\[
C = 180^\circ - 52^\circ - 68^\circ = 60^\circ
\]

Step 2: Use the Law of Sines to find \(AC\), the river width.

Let \(AC = b\), opposite angle \(B = 68^\circ\). Side \(AB = c = 150\) ft is opposite angle \(C = 60^\circ\).

\[
\frac{b}{\sin 68^\circ} = \frac{150}{\sin 60^\circ}
\]

Solve for \(b\):

\[
b = 150 \cdot \frac{\sin 68^\circ}{\sin 60^\circ}
\]

Calculator:

\[
b \approx 150 \cdot \frac{0.9272}{0.8660} \approx 160.6 \text{ ft}
\]

So the river is about 161 ft wide. This is a very practical example of solving problems using the Law of Sines, and it’s similar to what’s taught in many surveying and navigation courses.

For more background on triangle properties and trigonometry foundations, you can explore open course materials from places like MIT OpenCourseWare and Khan Academy, both widely used in U.S. classrooms.

When the Law of Cosines shines: Side‑angle‑side and side‑side‑side

The Law of Cosines is your go‑to when you either:

• Know two sides and the included angle (SAS), or
• Know all three sides (SSS) and want an angle.

The formula looks like a “Pythagorean theorem with a twist.” For triangle \(ABC\):

\[
a^2 = b^2 + c^2 - 2bc\cos A
\]

and similarly for \(b^2\) and \(c^2\).

Let’s walk through several examples of solving problems using the Law of Cosines in a way that feels more like story problems and less like abstract algebra.

Example of Law of Cosines: Finding a third side (SAS)

Suppose you have triangle \(ABC\) with:

• \(b = 7\) cm
• \(c = 10\) cm
• Included angle \(A = 120^\circ\)

Find side \(a\).

Step 1: Identify the pattern.
You know two sides and the angle between them (SAS). That’s perfect for the Law of Cosines.

Step 2: Plug into the formula.

\[
a^2 = b^2 + c^2 - 2bc\cos A
\]

\[
a^2 = 7^2 + 10^2 - 2(7)(10)\cos 120^\circ
\]

\[
a^2 = 49 + 100 - 140\cos 120^\circ
\]

Now, \(\cos 120^\circ = -\frac{1}{2}\).

\[
a^2 = 149 - 140\left(-\frac{1}{2}\right) = 149 + 70 = 219
\]

\[
a = \sqrt{219} \approx 14.8 \text{ cm}
\]

This is one of the best examples of how the Law of Cosines generalizes the Pythagorean theorem: if the angle were \(90^\circ\), the cosine term would vanish and you’d get the usual right‑triangle relation.

Real‑life style example: Distance between two hikers (Law of Cosines)

Two hikers start from the same campsite and walk in different directions:

• Hiker 1 walks 3 miles east.
• Hiker 2 walks 4 miles at a bearing that’s 50° north of east.

You’re asked: how far apart are they after they stop?

Draw triangle with the campsite as the shared starting point. The angle between their paths is \(50^\circ\), and the sides from the campsite are 3 and 4 miles.

Let \(d\) be the distance between the hikers.

Step 1: Recognize this as SAS.
Two sides and the included angle: 3 mi, 4 mi, and \(50^\circ\). Law of Cosines time.

\[
d^2 = 3^2 + 4^2 - 2(3)(4)\cos 50^\circ
\]

\[
d^2 = 9 + 16 - 24\cos 50^\circ
\]

Using \(\cos 50^\circ \approx 0.6428\):

\[
d^2 \approx 25 - 24(0.6428) \approx 25 - 15.4 = 9.6
\]

\[
d \approx \sqrt{9.6} \approx 3.1 \text{ miles}
\]

This is a neat example of solving problems using the Law of Cosines in a navigation‑flavored context, similar to how bearings and headings are treated in introductory physics and earth science classes.

If you’re curious how trigonometry shows up in navigation and Earth measurements, NASA’s education pages at nasa.gov often feature classroom activities that build on these same ideas.

Mixing both laws: Strategy examples of solving problems using the Law of Sines and Cosines

Some of the best examples in trigonometry combine both laws. Typically, you:

• Use the Law of Cosines first to get a side or angle, then
• Switch to the Law of Sines to finish the triangle.

Let’s walk through a couple of multi‑step examples.

Multi‑step example: From two sides and included angle to all angles

Triangle \(ABC\):

• \(b = 9\) m
• \(c = 12\) m
• Included angle \(A = 37^\circ\)

Find the remaining sides and angles.

Step 1: Use Law of Cosines to find side \(a\).

\[
a^2 = b^2 + c^2 - 2bc\cos A
\]

\[
a^2 = 9^2 + 12^2 - 2(9)(12)\cos 37^\circ
\]

\[
a^2 = 81 + 144 - 216\cos 37^\circ
\]

Using \(\cos 37^\circ \approx 0.7986\):

\[
a^2 \approx 225 - 216(0.7986) \approx 225 - 172.1 = 52.9
\]

\[
a \approx \sqrt{52.9} \approx 7.27 \text{ m}
\]

Step 2: Now use the Law of Sines to find an angle.
Let’s find angle \(B\).

\[
\frac{b}{\sin B} = \frac{a}{\sin A}
\]

\[
\frac{9}{\sin B} = \frac{7.27}{\sin 37^\circ}
\]

Solve for \(\sin B\):

\[
\sin B = 9 \cdot \frac{\sin 37^\circ}{7.27}
\]

Using \(\sin 37^\circ \approx 0.6018\):

\[
\sin B \approx 9 \cdot \frac{0.6018}{7.27} \approx 0.745
\]

\[
B \approx \sin^{-1}(0.745) \approx 48.1^\circ
\]

Step 3: Find the last angle using the angle sum.

\[
C = 180^\circ - A - B \approx 180^\circ - 37^\circ - 48.1^\circ \approx 94.9^\circ
\]

Here you’ve seen a full workflow example of solving problems using the Law of Sines and Cosines together: Cosines to break the SAS lock, then Sines to quickly get angles.

Ambiguous case example (SSA) with the Law of Sines

One of the most talked‑about examples of solving problems using the Law of Sines is the ambiguous case, when you know two sides and an angle not included between them (SSA).

Let’s say:

• \(a = 8\) cm
• \(b = 10\) cm
• Angle \(A = 30^\circ\)

Find angle \(B\), if possible.

Step 1: Apply the Law of Sines.

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

\[
\frac{8}{\sin 30^\circ} = \frac{10}{\sin B}
\]

\[
\frac{8}{0.5} = \frac{10}{\sin B} \Rightarrow 16 = \frac{10}{\sin B}
\]

\[
\sin B = \frac{10}{16} = 0.625
\]

Now, \(\sin B = 0.625\) could give two angles between 0° and 180°:

• \(B_1 \approx \sin^{-1}(0.625) \approx 38.7^\circ\)
• \(B_2 = 180^\circ - 38.7^\circ \approx 141.3^\circ\)

Step 2: Check which angles actually form a triangle.

If \(B = 141.3^\circ\), then

\[
A + B = 30^\circ + 141.3^\circ = 171.3^\circ
\]

leaving only \(8.7^\circ\) for angle \(C\). That still works, so two triangles are possible here.

This is a textbook example of the ambiguous case, widely discussed in modern high school curricula. The Common Core State Standards for high school geometry in the U.S. explicitly mention using trigonometric ratios to solve real‑world and mathematical problems, and this SSA situation is a favorite test topic.

2024‑style applications: Data, drones, and design

Trigonometry hasn’t changed, but how we use it has evolved with technology. Some of the best examples of solving problems using the Law of Sines and Cosines now show up in:

• Drone flight and mapping – Estimating distances and angles between waypoints when the drone’s path isn’t a simple right angle.
• Computer graphics and game development – Computing distances between points and angles between vectors, which often boil down to Law of Cosines‑style calculations.
• Architecture and structural engineering – Working with non‑right triangles in roof trusses, bridges, and support structures.

For instance, imagine a drone flying from point A to B, then turning and flying to C, with its GPS giving you the lengths of AB and AC and the turning angle at A. Estimating the straight‑line distance from B to C is another real example of solving problems using the Law of Cosines.

If you’re interested in STEM career pathways where these skills show up, the U.S. Bureau of Labor Statistics at bls.gov outlines math‑heavy careers that rely on this kind of geometric thinking.

Quick strategy guide for choosing Sine vs. Cosine

When you face a new triangle, pause and sort it into a pattern. Some of the best examples of exam success come from just recognizing the setup:

• AAS or ASA (two angles and any side): Law of Sines is usually faster.
• SAS (two sides with the included angle): Start with Law of Cosines.
• SSS (all three sides): Use Law of Cosines to get one angle, then Law of Sines if needed.
• SSA (two sides and a non‑included angle): Law of Sines, but watch for the ambiguous case (0, 1, or 2 solutions).

If you consistently label your triangle (angle A opposite side a, etc.) and write down what you know in pairs (side + opposite angle), you’ll quickly see whether the examples of solving problems using the Law of Sines and Cosines you’ve practiced match the situation in front of you.

FAQ: Common questions about examples of solving problems using the Law of Sines and Cosines

Q: Can you give a simple example of when to use the Law of Sines instead of the Law of Cosines?
If you know two angles and one side (say, angles A and B, and side a), that’s a natural Law of Sines situation. You can quickly find the third angle using angle sum, then use side‑angle pairs in the Law of Sines to get the remaining sides.

Q: What are some real examples of the Law of Cosines in everyday life?
Any time you’re dealing with distances that don’t meet at right angles. Examples include the distance between two roads leaving a junction at an angle, the span of a bridge between supports that aren’t aligned straight across, or the distance between two aircraft flying on different headings.

Q: Is there an example of using both laws in one problem that’s especially useful for exams?
Yes. A favorite exam pattern is SAS: you’re given two sides and the included angle, asked to find everything else. You use the Law of Cosines once to get the third side, then switch to the Law of Sines to get one of the remaining angles, and finally use angle sum for the last angle.

Q: How accurate do my answers need to be in these examples of triangle problems?
That depends on your teacher or exam instructions, but most math classes accept answers rounded to two or three decimal places for side lengths and one decimal place for angles. In real‑world engineering or physics, the required precision depends on the application.

Q: Where can I practice more examples of solving problems using the Law of Sines and Cosines?
Many U.S. high school and college courses provide free practice. You can check open resources from universities like Harvard’s math resources or browse problem sets on educational platforms that focus on trigonometry and geometry.

The more you work through these step‑by‑step examples of solving problems using the Law of Sines and Cosines, the more they start to feel like familiar patterns instead of puzzles. With a bit of organized practice, you’ll reach the point where you can look at a triangle, recognize the setup, and know exactly which law to reach for next.