Sine Rule, Cosine Rule and Projection Formula | NIMCET

Quick answer: In triangle ABC with sides $a, b, c$ opposite angles $A, B, C$, the sine rule is $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$, the cosine rule is $a^2 = b^2 + c^2 - 2bc\cos A$, and the projection formula is $a = b\cos C + c\cos B$. Use the sine rule when you know an angle and its opposite side; use the cosine rule when you know two sides and the included angle, or all three sides.

Notation

Throughout, $A + B + C = \pi$, which links this topic to the conditional identities.

Sine rule

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

So $a = 2R\sin A$, $b = 2R\sin B$, $c = 2R\sin C$. This lets you swap sides for sines in any identity, which is the key move in many proofs.

Consequence: the larger side is opposite the larger angle.

Cosine rule

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

Rearranged to find an angle:

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

The other two follow by symmetry: $\cos B = \frac{c^2 + a^2 - b^2}{2ca}$ and $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$.

Angle type test: if $a^2 > b^2 + c^2$, angle A is obtuse; if equal, it is a right angle; if less, it is acute.

Projection formulas

Each side equals the sum of the projections of the other two sides on it.

Napier's analogy (tangent rule)

$\tan\frac{B - C}{2} = \frac{b - c}{b + c}\cot\frac{A}{2}$

This is useful when two sides and the included angle are known and you want the difference of the other two angles.

Which rule to use

The ambiguous case

Given $a$, $b$ and angle $A$ (acute), the number of triangles depends on $b\sin A$:

Worked examples

Example 1: finding an angle from three sides

Sides are 3, 5 and 7. Find the largest angle.

The largest angle is opposite 7:

$\cos C = \frac{9 + 25 - 49}{2 \cdot 3 \cdot 5} = \frac{-15}{30} = -\frac{1}{2}$

So $C = 120^\circ$.

Example 2: sine rule

In triangle ABC, $A = 45^\circ$, $B = 60^\circ$ and $a = \sqrt{2}$. Find $b$.

$b = \frac{a\sin B}{\sin A} = \frac{\sqrt{2} \cdot \frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}} = \sqrt{3}$

Example 3: using a = 2R sin A in a proof

Show that $a\cos A + b\cos B + c\cos C = 4R\sin A\sin B\sin C$.

Step 1: Replace each side: the left side becomes $R(\sin 2A + \sin 2B + \sin 2C)$.

Step 2: Since $A + B + C = \pi$, $\sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C$, which gives the result.

Example 4: angle type

Is the triangle with sides 4, 5 and 7 acute, right or obtuse?

$7^2 = 49 > 4^2 + 5^2 = 41$, so the angle opposite 7 is obtuse.

Summary table

Exam tips

• In identity proofs, convert sides to sines with $a = 2R\sin A$, or sines to sides with $\sin A = \frac{a}{2R}$.
• Find the largest angle from the largest side; it is the only one that can be obtuse.
• Remember the 3, 5, 7 triangle: its largest angle is exactly 120°.
• In the ambiguous case, always compare $a$ with $b\sin A$ and $b$.

For more chapter topics, visit the NIMCET Trigonometry chapter.

Common mistakes

• Pairing a side with the wrong angle; side $a$ is always opposite angle $A$.
• Forgetting the minus sign in the cosine rule.
• Missing the second triangle in the ambiguous case.
• Using the sine rule to find an obtuse angle without checking: $\sin\theta$ is the same for θ and 180° − θ.

Frequently Asked Questions

What is the sine rule?

In any triangle, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

What is the cosine rule?

a² = b² + c² − 2bc cos A, or equivalently cos A = (b² + c² − a²)/(2bc).

When should I use the cosine rule instead of the sine rule?

Use the cosine rule when you know all three sides, or two sides and the angle between them.

What is the projection formula?

a = b cos C + c cos B. Each side equals the sum of the projections of the other two sides on it.

What is the largest angle of a triangle with sides 3, 5 and 7?

It is 120°, since cos C = (9 + 25 − 49)/30 = −1/2.