From “Why Are We Learning This?” to “What’s the Next Problem?”

A Precalculus Classroom Story About the Law of Sines and Law of Cosines

One of the questions I hear most often in math class is:

"Why are we learning this?"

It is a fair question. Students want to know how the concepts they learn connect to the world around them. Recently, during an online Precalculus class, that question led to one of the most engaging lessons I have taught.

Instead of opening another slide filled with formulas, I paused and shared a simple story.

A Camping Trip and a Triangle

I asked my students to imagine three friends camping in the woods.

Their tents formed a triangle.

• Priya and Rohan were 10 metres apart.

• The angle at Priya’s tent was 30°.

• The angle at Arjun’s tent was 105°.

The challenge was simple:

How far apart are Rohan and Arjun?

Suddenly, the lesson felt different.

Instead of looking at letters and formulas on a page, students were solving a real problem. We drew the triangle together and discussed what information we already knew.

This was the perfect opportunity to introduce the Law of Sines.

Law of Sines

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

The Law of Sines helps us find unknown sides or angles in a triangle when we know:

• Two angles and one side (ASA or AAS)

• Two sides and a non-included angle (SSA)

As we worked through the problem, one student suddenly realized something interesting.

"Ma’am, we just found a distance nobody actually measured!"

Exactly.

That moment captured the true power of mathematics. We can use logical relationships to discover information that is not directly visible.

A Lake, a Surveyor, and Another Problem

After solving the camping problem, we moved on to a different situation.

Imagine a surveyor standing beside a lake. They need to determine the distance across the water, but crossing the lake is not practical.

This time, we knew:

• Two sides of a triangle

• The angle between those sides

This was the perfect setting to introduce the Law of Cosines.

Law of Cosines

c^2=a^2+b^2-2abcos C

The Law of Cosines is useful when we know:

• Two sides and the included angle (SAS)

• All three sides of a triangle (SSS)

Students quickly noticed an important idea:

Different triangles require different strategies.

Just as different tools are needed for different tasks, different triangle situations call for different mathematical approaches.

More Than Just Formulas

As the lesson continued, our discussion expanded beyond camping trips and lakes.

We talked about:

• Airplane navigation

• Satellite positioning

• Surveying land

• Engineering and construction

• Mapping and GPS technology

Students began to see that trigonometry is not simply a collection of formulas to memorize. It is a powerful way of solving real-world problems.

Mathematics helps us:

• Analyze situations logically

• Make predictions using available information

• Solve problems that cannot be measured directly

• Connect abstract ideas to practical applications

The Most Important Lesson

I often tell my students:

The formula is the easy part. Knowing when to use it is the real skill.

Understanding mathematical concepts is not about memorizing steps. It is about recognizing patterns, choosing the right approach, and applying knowledge to new situations.

By the end of class, nobody was asking:

"Why are we learning this?"

Instead, students were asking:

"Ma’am, what’s the next problem?"

That shift—from confusion to curiosity—is one of the most rewarding moments in teaching.

Final Thoughts

Mathematics becomes meaningful when students can connect it to their everyday experiences. Whether it is measuring distances between camping tents, calculating across a lake, navigating an aircraft, or designing structures, the principles of trigonometry help us understand and solve real problems.

As an online math tutor, I have learned that students become more engaged when they see the purpose behind what they are learning. Sometimes all it takes is one relatable story to transform a lesson from a formula on a page into an exciting problem-solving adventure.

Supriya SumanOnline Math Tutor