The Cosine Rule: Its easier than it seems

What is the Cosine Rule?

The main formula

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

The Cosine Rule is mainly used to solve non-right-angled triangles.

The lowercase side a must be opposite the capital angle A. The same matching rule works for any other side and angle.

When students first see the Cosine Rule, many think the formula looks difficult.

But the process is actually very repetitive and structured.

Every question follows the same idea:

\[ \text{identify the sides} \rightarrow \text{substitute carefully} \rightarrow \text{solve step-by-step} \]

The most important step is matching each angle with the side directly opposite it.

Sides next to angle A Side a, opposite angle A Angle A

Side \(a\) is opposite angle \(A\).

1. When should we use the Cosine Rule?

The Cosine Rule is used in non-right-angled triangles.

It is especially useful in two situations:

when we know two sides and the angle between them;
when we know all three sides and need an angle.

Choose the rule from the information given

Use the Cosine Rule for two sides and the included angle (SAS).
Use the Cosine Rule for all three sides when you need an angle (SSS).
The Sine Rule is often better when you know a matching side-and-opposite-angle pair.

2. Understanding the formula

The Cosine Rule is:

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

At first glance, this formula looks complicated.

But the structure is actually simple.

\[ \text{side squared} = \text{side squared} + \text{side squared} – \text{adjustment} \]

The cosine part changes the answer depending on the size of the angle.

With the two surrounding sides fixed, a smaller angle gives a shorter opposite side.
With the two surrounding sides fixed, a larger angle gives a longer opposite side.

3. Example — finding a missing side

Question

Find the missing side \(x\).

Step 1 — Identify what we know

We know:

\[ 5\text{ cm}, \quad 8\text{ cm}, \quad 60^\circ \]

The angle is between the two known sides.

That means this is a Cosine Rule question.

We are trying to find the side opposite the angle.

\[ a=x \]

Step 2 — Write the formula

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

Step 3 — Substitute values carefully

Replace each letter with the correct number.

\[ x^2=5^2+8^2-2(5)(8)\cos(60^\circ) \]

Step 4 — Calculate powers first

\[ x^2=25+64-80\cos(60^\circ) \]

Now evaluate the cosine.

\[ \cos(60^\circ)=0.5 \]

\[ x^2=25+64-80(0.5) \]

Step 5 — Simplify

\[ x^2=89-40 \]

\[ x^2=49 \]

Step 6 — Square root both sides

We need \(x\), not \(x^2\), so we square root both sides.

\[ x=\sqrt{49} \]

\[ x=7 \]

\[ \boxed{x=7\text{ cm}} \]

Always remember:

If the formula gives \(x^2\), the final step is square rooting.

4. Example — finding a missing angle

Question

Find angle \(A\). Give your answer to the nearest degree.

Step 1 — Identify the opposite side

The angle we need is \(A\).

The side opposite angle \(A\) is:

\[ 11\text{ cm} \]

So:

\[ a=11 \]

The remaining sides are:

\[ b=6,\quad c=9 \]

Step 2 — Rearranged formula

When finding an angle, it is easier to use:

\[ \cos(A)=\frac{b^2+c^2-a^2}{2bc} \]

Step 3 — Substitute values

\[ \cos(A)=\frac{6^2+9^2-11^2}{2(6)(9)} \]

Step 4 — Evaluate powers

\[ \cos(A)=\frac{36+81-121}{108} \]

\[ \cos(A)=\frac{-4}{108} \]

\[ \cos(A)\approx -0.0370 \]

Keep the full calculator value until the final answer to avoid rounding errors.

Step 5 — Use inverse cosine

At the moment we have the cosine value, but we need the angle itself.

So we use inverse cosine.

\[ A=\cos^{-1}(-0.037) \]

\[ A\approx92.1^\circ \]

\[ \boxed{A\approx92^\circ} \]

When finding an angle:

\[ \cos^{-1} \]

is one of the most important calculator buttons.

5. Exercises

Exercise 1

Find the missing side \(x\). Give your answer to 1 decimal place.

Show explanation and answer

Explanation and Answer

We know two sides and the angle between them.

That means we use the Cosine Rule.

The side we are finding is opposite the angle.

\[ a=x,\quad b=7,\quad c=10 \]

Write the formula:

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

Substitute:

\[ x^2=7^2+10^2-2(7)(10)\cos(45^\circ) \]

Calculate powers:

\[ x^2=49+100-140\cos(45^\circ) \]

Now evaluate the cosine:

\[ \cos(45^\circ)\approx0.7071 \]

\[ x^2=149-140(0.7071) \]

\[ x^2=149-98.99 \]

\[ x^2\approx50.01 \]

Square root both sides:

\[ x\approx\sqrt{50.01} \]

\[ x\approx7.1 \]

\[ \boxed{x\approx7.1\text{ cm}} \]

Exercise 2

Find the missing side \(x\). Give your answer to 1 decimal place.

Show explanation and answer

Explanation and Answer

We know:

\[ 6\text{ cm}, \quad 9\text{ cm}, \quad 110^\circ \]

The angle is between the known sides, so we use the Cosine Rule.

\[ x^2=6^2+9^2-2(6)(9)\cos(110^\circ) \]

Calculate powers:

\[ x^2=36+81-108\cos(110^\circ) \]

Now evaluate the cosine.

\[ \cos(110^\circ)\approx -0.342 \]

Notice that the cosine is negative because the angle is obtuse.

\[ x^2=117-108(-0.342) \]

Subtracting a negative becomes addition.

\[ x^2=117+36.94 \]

\[ x^2\approx153.94 \]

Now square root both sides:

\[ x\approx\sqrt{153.94} \]

\[ x\approx12.4 \]

\[ \boxed{x\approx12.4\text{ cm}} \]

Exercise 3

Find angle \(A\). Give your answer to the nearest degree.

Show explanation and answer

Explanation and Answer

All three sides are known.

That means we use the Cosine Rule to find an angle.

The side opposite angle \(A\) is:

\[ 7\text{ cm} \]

So:

\[ a=7,\quad b=8,\quad c=10 \]

Use the rearranged formula:

\[ \cos(A)=\frac{b^2+c^2-a^2}{2bc} \]

Substitute values:

\[ \cos(A)=\frac{8^2+10^2-7^2}{2(8)(10)} \]

Evaluate powers:

\[ \cos(A)=\frac{64+100-49}{160} \]

\[ \cos(A)=\frac{115}{160} \]

\[ \cos(A)=0.71875 \]

Now use inverse cosine:

\[ A=\cos^{-1}(0.71875) \]

\[ A\approx44^\circ \]

\[ \boxed{A\approx44^\circ} \]

Exercise 4

A triangular garden has sides of length \(12\) m and \(15\) m.

The angle between them is \(70^\circ\).

Find the third side.

Give your answer to 1 decimal place.

Show explanation and answer

Explanation and Answer

This is a real-life problem, but mathematically it is still a Cosine Rule question.

We know two sides and the included angle.

Write the formula:

\[ x^2=12^2+15^2-2(12)(15)\cos(70^\circ) \]

Calculate powers:

\[ x^2=144+225-360\cos(70^\circ) \]

Evaluate the cosine:

\[ \cos(70^\circ)\approx0.342 \]

Substitute:

\[ x^2=369-360(0.342) \]

\[ x^2=369-123.12 \]

\[ x^2\approx245.88 \]

Square root:

\[ x\approx\sqrt{245.88} \]

\[ x\approx15.7 \]

\[ \boxed{x\approx15.7\text{ m}} \]

Exercise 5

A triangle has side lengths:

\[ 9\text{ cm},\quad 11\text{ cm},\quad 14\text{ cm} \]

Find the angle opposite the side of length \(14\) cm.

Give your answer to the nearest degree.

Show explanation and answer

Explanation and Answer

The angle we need is opposite the side of length \(14\) cm.

So:

\[ a=14,\quad b=9,\quad c=11 \]

Use the rearranged formula:

\[ \cos(A)=\frac{b^2+c^2-a^2}{2bc} \]

Substitute:

\[ \cos(A)=\frac{9^2+11^2-14^2}{2(9)(11)} \]

Evaluate powers:

\[ \cos(A)=\frac{81+121-196}{198} \]

\[ \cos(A)=\frac{6}{198} \]

\[ \cos(A)\approx0.0303 \]

Now use inverse cosine:

\[ A=\cos^{-1}(0.0303) \]

\[ A\approx88^\circ \]

\[ \boxed{A\approx88^\circ} \]

Because the cosine value is very close to zero, the angle is very close to \(90^\circ\).

Exercise 6 — Real Life Application

A rescue team is planning a zipline across a canyon.

One side of the canyon is connected to a point that is \(120\) m away from the launch platform.

Another support point is \(85\) m away from the same platform.

The angle between the two support cables is \(48^\circ\).

Find the distance between the two support points. Give your answer to 1 decimal place.

Show explanation and answer

Explanation and Answer

This is a real-life problem involving distances and an angle.

We know:

\[ 120\text{ m}, \quad 85\text{ m}, \quad 48^\circ \]

The angle is between the two known sides.

That means this is a Cosine Rule question.

We are trying to find the side opposite the angle.

\[ a=x,\quad b=120,\quad c=85 \]

Write the Cosine Rule:

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

Substitute the values:

\[ x^2=120^2+85^2-2(120)(85)\cos(48^\circ) \]

Calculate the powers first:

\[ x^2=14400+7225-20400\cos(48^\circ) \]

Now evaluate the cosine:

\[ \cos(48^\circ)\approx0.6691 \]

Substitute this value:

\[ x^2=21625-20400(0.6691) \]

Multiply carefully:

\[ x^2=21625-13649.64 \]

\[ x^2\approx7975.36 \]

Now square root both sides to find \(x\):

\[ x\approx\sqrt{7975.36} \]

\[ x\approx89.3 \]

\[ \boxed{x\approx89.3\text{ m}} \]

Notice how this real-life problem still follows exactly the same steps:

\[ \text{identify the sides} \rightarrow \text{substitute carefully} \rightarrow \text{calculate step-by-step} \]

The context changes, but the mathematics stays the same.

6. Final summary

1Identify the opposite side carefully.

2Match the angle with the correct side letter.

3Substitute slowly and carefully.

4Use brackets properly.

5Use \(\cos^{-1}\) for missing angles.

6Square root at the end for missing sides.

\[ \boxed{ \text{match the sides} \rightarrow \text{substitute carefully} \rightarrow \text{solve step-by-step} } \]

Final Checklist

Is the calculator in degree mode?
Did you identify the opposite side correctly?
Did you use inverse cosine for angles?
Did you square root at the end for sides?