Trigonometry — sine and cosine rules (KCSE Mathematics Form 4)

TL;DR

The sine rule helps you find unknown sides or angles in any triangle when you have certain pairs of information. The cosine rule is used when the sine rule doesn't apply, typically with three sides and one angle. Together, these rules let you solve for missing parts of non-right-angled triangles.

1. The Mental Model

Imagine you have a triangle that isn't a right-angled one. You can't use SOH CAH TOA directly. The sine and cosine rules are your special tools to figure out its missing sides or angles.

2. The Core Material

When you're dealing with triangles that don't have a 90-degree angle, the sine rule and cosine rule become super important. Let's break them down.

2.1 The Sine Rule

The sine rule is used when you have a pair of an angle and its opposite side, plus one other piece of information (either another angle or another side).

Formula:
For any triangle ABC, with sides a, b, c opposite to angles A, B, C respectively:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

You can also write it as:
$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

When to use it:
* Finding a side: If you know two angles and one side (AAS or ASA).
* Finding an angle: If you know two sides and an angle opposite one of those sides (SSA). Be careful with SSA, as it can sometimes lead to two possible triangles (the ambiguous case), but for KCSE, you'll usually be guided to the correct one or it will be obvious.

Example: Finding a side
Suppose you have a triangle ABC where angle A = 40°, angle B = 60°, and side a = 10 cm. Find side b.

Using the sine rule:
$\frac{a}{\sin A} = \frac{b}{\sin B}$
$\frac{10}{\sin 40^\circ} = \frac{b}{\sin 60^\circ}$
$b = \frac{10 \times \sin 60^\circ}{\sin 40^\circ}$
$b = \frac{10 \times 0.8660}{0.6428}$
$b \approx 13.47 \text{ cm}$

Example: Finding an angle
Suppose you have a triangle ABC where side a = 12 cm, side b = 15 cm, and angle A = 50°. Find angle B.

Using the sine rule:
$\frac{\sin A}{a} = \frac{\sin B}{b}$
$\frac{\sin 50^\circ}{12} = \frac{\sin B}{15}$
$\sin B = \frac{15 \times \sin 50^\circ}{12}$
$\sin B = \frac{15 \times 0.7660}{12}$
$\sin B = 0.9575$
$B = \sin^{-1}(0.9575)$
$B \approx 73.2^\circ$

2.2 The Cosine Rule

The cosine rule is more versatile and can be used when the sine rule doesn't give you enough information. It relates the lengths of the sides of a triangle to the cosine of one of its angles.

Formula:
For any triangle ABC, with sides a, b, c opposite to angles A, B, C respectively:
* $a^2 = b^2 + c^2 - 2bc \cos A$
* $b^2 = a^2 + c^2 - 2ac \cos B$
* $c^2 = a^2 + b^2 - 2ab \cos C$

You can also rearrange these to find an angle:
* $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
* $\cos B = \frac{a^2 + c^2 - b^2}{2ac}$
* $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

When to use it:
* Finding a side: If you know two sides and the included angle (SAS). The included angle is the angle between the two known sides.
* Finding an angle: If you know all three sides (SSS).

Example: Finding a side
Suppose you have a triangle ABC where side b = 8 cm, side c = 10 cm, and angle A = 60°. Find side a.

Using the cosine rule:
$a^2 = b^2 + c^2 - 2bc \cos A$
$a^2 = 8^2 + 10^2 - 2(8)(10) \cos 60^\circ$
$a^2 = 64 + 100 - 160 \times 0.5$
$a^2 = 164 - 80$
$a^2 = 84$
$a = \sqrt{84}$
$a \approx 9.17 \text{ cm}$

Example: Finding an angle
Suppose you have a triangle ABC where side a = 7 cm, side b = 9 cm, and side c = 11 cm. Find angle C.

Using the rearranged cosine rule:
$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$
$\cos C = \frac{7^2 + 9^2 - 11^2}{2(7)(9)}$
$\cos C = \frac{49 + 81 - 121}{126}$
$\cos C = \frac{9}{126}$
$\cos C = 0.0714$
$C = \cos^{-1}(0.0714)$
$C \approx 85.9^\circ$

2.3 Deciding Which Rule to Use

It's crucial to know when to apply which rule. Here's a simple decision flow:

graph TD
    A[Start] --> B{Is it a right-angled triangle?};
    B -- Yes --> C[Use SOH CAH TOA / Pythagoras Theorem];
    B -- No --> D{Do you have a side and its opposite angle?};
    D -- Yes --> E[Use Sine Rule];
    D -- No --> F{Do you have two sides and the included angle (SAS)?};
    F -- Yes --> G[Use Cosine Rule to find the third side];
    F -- No --> H{Do you have all three sides (SSS)?};
    H -- Yes --> I[Use Cosine Rule to find any angle];
    H -- No --> J[You might be missing information or need to find another angle first];

3. Worked Example

Problem:
A triangle PQR has sides p = 15 cm, q = 18 cm, and angle R = 70°.
a) Calculate the length of side r.
b) Calculate the size of angle P.

Solution:

a) Calculate the length of side r.
We have two sides (p and q) and the included angle (R). This is a SAS case, so we use the cosine rule to find side r.

Using the cosine rule:
$r^2 = p^2 + q^2 - 2pq \cos R$
$r^2 = 15^2 + 18^2 - 2(15)(18) \cos 70^\circ$
$r^2 = 225 + 324 - 540 \times 0.3420$ (using $\cos 70^\circ \approx 0.3420$)
$r^2 = 549 - 184.68$
$r^2 = 364.32$
$r = \sqrt{364.32}$
$r \approx 19.09 \text{ cm}$ (to 2 decimal places)

b) Calculate the size of angle P.
Now we know all three sides (p=15, q=18, r=19.09) and one angle (R=70°). We can use either the sine rule or the cosine rule. The sine rule is often quicker if you have a side-angle pair. We have r and R, and we want to find angle P, opposite side p.

Using the sine rule:
$\frac{\sin P}{p} = \frac{\sin R}{r}$
$\frac{\sin P}{15} = \frac{\sin 70^\circ}{19.09}$
$\sin P = \frac{15 \times \sin 70^\circ}{19.09}$
$\sin P = \frac{15 \times 0.9397}{19.09}$ (using $\sin 70^\circ \approx 0.9397$)
$\sin P = \frac{14.0955}{19.09}$
$\sin P \approx 0.7384$
$P = \sin^{-1}(0.7384)$
$P \approx 47.6^\circ$ (to 1 decimal place)

(Alternatively, using the cosine rule for angle P:)
$\cos P = \frac{q^2 + r^2 - p^2}{2qr}$
$\cos P = \frac{18^2 + 19.09^2 - 15^2}{2(18)(19.09)}$
$\cos P = \frac{324 + 364.4381 - 225}{687.24}$
$\cos P = \frac{463.4381}{687.24}$
$\cos P \approx 0.6743$
$P = \cos^{-1}(0.6743)$
$P \approx 47.6^\circ$ (to 1 decimal place)

Both methods give the same answer, which is a good sign!

4. Key Takeaways

• The sine rule is for when you have a side and its opposite angle, plus one other piece of information.
• The cosine rule is for when you have two sides and the included angle (SAS) or all three sides (SSS).
• Always draw a diagram of the triangle and label it correctly before starting.
• Remember that angles in a triangle add up to 180 degrees; this can