Year 11 General Applications Of Trigonometry

The Cosine Rule

Q: What is the cosine rule?

The cosine rule for a side: c squared equals a squared plus b squared minus 2 a b cos C, where a and b are two sides of a triangle, C is the angle between them, and c is the side opposite C. It works for any triangle and generalises Pythagoras' theorem (which is just the special case C equals 90 degrees).

Q: When do I use the cosine rule?

Use the cosine rule when the sine rule can't help: either SAS (two sides and the included angle, find the third side) or SSS (all three sides, find any angle). If you have a matched side-and-opposite-angle pair, the sine rule is faster.

Q: How is the cosine rule related to Pythagoras?

When C equals 90 degrees, cos C equals 0, so the cosine rule reduces to c squared equals a squared plus b squared — Pythagoras' theorem. The cosine rule is the same idea generalised to any triangle, with the correction term minus 2 a b cos C accounting for non-right angles.

Q: How do I rearrange the cosine rule to find an angle?

Make the cosine the subject: cos C equals (a squared plus b squared minus c squared) divided by (2 a b). The angle you're finding is opposite the side that's squared with the minus sign on the right. Then apply cos inverse on your calculator.

Q: Why is the cosine negative for an obtuse angle?

Cosine is negative for angles between 90 degrees and 180 degrees. So if your cos C calculation gives a negative value, C is obtuse — this is correct, not a mistake. The side opposite an obtuse angle is always the longest in the triangle.

Q: What's the difference between SAS and SSA?

SAS (side-angle-side): two sides and the angle between them. Use the cosine rule. SSA (side-side-angle): two sides and an angle not between them. Use the sine rule — but watch out for the ambiguous case.

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Theory

The cosine rule is a generalisation of Pythagoras' theorem to any triangle: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$ . Use it whenever the sine rule can't — when you have SAS (two sides and the included angle) or SSS (all three sides).

The cosine rule is a generalisation of Pythagoras' theorem to any triangle. It picks up where the sine rule can't help — when you don't have a matched side-and-opposite-angle pair.

To find a side: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$ , where $a$ and $b$ are the two sides bracketing angle $C$ , and $c$ is the side opposite $C$ .
To find an angle: rearrange to make the cosine the subject: $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$ . The angle on the left is the one opposite the side that's squared on the right.

Two situations where the cosine rule is needed:

SAS — given two sides and the included angle. Use the first form to find the third side.
SSS — given all three sides. Use the second form to find any angle.

Connection to Pythagoras: if $C = 90^{\circ}$ , then $\cos C = 0$ , and the formula collapses to $c^{2} = a^{2} + b^{2}$ — Pythagoras' theorem. The cosine rule is the same idea with a correction term $- 2 a b \cos C$ for non-right angles.

SAS:

a

and

b

bracket angle

C

;

c

is opposite. Use

c^{2} = a^{2} + b^{2} - 2 a b \cos C

SSS: largest angle is opposite the longest side

Cosine rule — finding a side (SAS)

c^{2} = a^{2} + b^{2} - 2 a b \cos C

c^{2} = a^{2} + b^{2} - 2 a b \cos C

Cosine rule — finding an angle (SSS)

Rearrange to make the cosine the subject. The angle on the left is the one opposite the side that's squared with a minus sign on the right:

\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}

\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}

Connection to Pythagoras

When $C = 90^{\circ}$ , $\cos C = 0$ , so the cosine rule reduces to:

c^{2} = a^{2} + b^{2} (Pythagoras)

c^{2} = a^{2} + b^{2}

          Combined sine + cosine: many problems use the cosine rule first to find one missing side or angle, then the sine rule for any further unknowns. Use the cosine rule sparingly — it's slower than the sine rule when both work.
        

How to use the cosine rule

Label the triangle: each side has the lowercase of its opposite vertex.
Identify the situation — SAS (two sides + included angle) calls for the first form; SSS (three sides) calls for the rearranged form.
Substitute carefully — for SSS, the angle on the left is the one opposite the squared side with the minus sign.
Solve — take the square root (positive only) for a side, or $\cos^{- 1}$ for an angle.

Example 1 — SAS, find a side

In triangle

A B C

A B = 8

cm,

A C = 11

cm, and

∠ A = 62^{\circ}

. Find

B C

(1 dp).

Solution

Sides $8$ and $11$ bracket angle $A = 62^{\circ}$ , so $B C = a$ :

$a^{2}$	$=$	$8^{2} + 11^{2} - 2 \cdot 8 \cdot 11 \cdot \cos 62^{\circ}$
$a^{2}$	$=$	$64 + 121 - 176 \cos 62^{\circ}$
$a^{2}$	$\approx$	$102.36$
$a$	$\approx$	$10.1 cm$

a = \sqrt{102.36} \approx 10.1 cm

Example 2 — SSS, find an angle

In triangle

P Q R

p = 7

q = 9

r = 12

. Find

∠ P

(nearest degree).

Solution

$∠ P$ is opposite $p = 7$ , so put $p^{2}$ with the minus sign on top:

$\cos P$	$=$	$\frac{q^{2} + r^{2} - p^{2}}{2 q r}$
$\cos P$	$=$	$\frac{81 + 144 - 49}{216}$
$\cos P$	$=$	$\frac{176}{216} \approx 0.8148$
$P$	$\approx$	$35^{\circ}$

P = \cos^{- 1} (0.8148) \approx 35 °

Example 3 — Largest angle

A triangle has sides

7

9

, and

13

cm. Find the largest angle (nearest degree).

Solution

The largest angle is opposite the longest side $13$ , so put $13^{2}$ with the minus sign on top:

$\cos θ$	$=$	$\frac{7^{2} + 9^{2} - 13^{2}}{2 \cdot 7 \cdot 9}$
$\cos θ$	$=$	$\frac{49 + 81 - 169}{126}$
$\cos θ$	$=$	$\frac{- 39}{126} \approx - 0.3095$
$θ$	$\approx$	$108^{\circ}$

The negative cosine confirms the angle is obtuse.

θ = \cos^{- 1} (- 0.3095) \approx 108 °

Example 4 — Bearing problem

A ship sails from port

A

on bearing

075^{\circ}

for

45

km to

B

, then on bearing

140^{\circ}

for

60

km to

C

. The angle

∠ A B C = 115^{\circ}

. Find

A C

(1 dp).

Solution

Sketch — sides $A B = 45$ and $B C = 60$ bracket the angle at $B$ :

$A C^{2}$	$=$	$45^{2} + 60^{2} - 2 (45) (60) \cos 115^{\circ}$
$A C^{2}$	$=$	$2025 + 3600 - 5400 \cos 115^{\circ}$
$A C^{2}$	$\approx$	$7907$
$A C$	$\approx$	$88.9 km$

A C = \sqrt{7907} \approx 88.9 km

Common pitfalls

The angle must be the included angle. The angle in

\cos C

must be the angle between sides

a

and

b

. Using a non-included angle gives the wrong answer.

Obtuse angles give negative cosines — that's correct. If

C > 90^{\circ}

\cos C

is negative, so

- 2 a b \cos C

is positive, making

c^{2}

larger. The side opposite an obtuse angle is always the longest.

SSS: largest angle opposite longest side. If you want the largest angle quickly, put the longest side as

c

in the rearranged formula. A negative cosine confirms it's obtuse.

Don't use cosine rule when sine rule works. If you have a matched side-and-opposite-angle pair, the sine rule is faster. Use the cosine rule only when the sine rule can't help (SAS or SSS).

Frequently asked questions

What is the cosine rule?

The cosine rule for a side: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$ , where $a$ and $b$ are two sides of a triangle, $C$ is the angle between them, and $c$ is the side opposite $C$ . It works for any triangle and generalises Pythagoras' theorem (which is just the special case $C = 90^{\circ}$ ).

When do I use the cosine rule?

Use the cosine rule when the sine rule can't help: either SAS (two sides and the included angle, find the third side) or SSS (all three sides, find any angle). If you have a matched side-and-opposite-angle pair, the sine rule is faster.

How is the cosine rule related to Pythagoras?

When $C = 90^{\circ}$ , $\cos C = 0$ , so the cosine rule reduces to $c^{2} = a^{2} + b^{2}$ — Pythagoras' theorem. The cosine rule is the same idea generalised to any triangle, with the correction term $- 2 a b \cos C$ accounting for non-right angles.

How do I rearrange the cosine rule to find an angle?

Make the cosine the subject: $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$ . The angle you're finding is opposite the side that's squared with the minus sign on the right. Then apply $\cos^{- 1}$ on your calculator.

Why is the cosine negative for an obtuse angle?

Cosine is negative for angles between $90^{\circ}$ and $180^{\circ}$ . So if your $\cos C$ calculation gives a negative value, $C$ is obtuse — this is correct, not a mistake. The side opposite an obtuse angle is always the longest in the triangle.

What's the difference between SAS and SSA?

SAS (side-angle-side): two sides and the angle between them. Use the cosine rule. SSA (side-side-angle): two sides and an angle not between them. Use the sine rule — but watch out for the ambiguous case.

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The Cosine Rule

📖 Theory

Cosine rule — finding a side (SAS)

Cosine rule — finding an angle (SSS)

Connection to Pythagoras

How to use the cosine rule

Common pitfalls

Frequently asked questions

Theory