Sine and Cosine Rules

The Sine Rule

Think of the Sine Rule as a set of perfectly balanced scales. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of a triangle.

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

To find a missing side, use the standard formula. To find a missing angle, flip the fractions upside down ($\frac{\sin A}{a} = \frac{\sin B}{b}$) so the unknown is on top, making it much easier to rearrange. You will then need to use the inverse sine ($\sin^{-1}$) function on your calculator to find the final angle.

Watch out for the ambiguous case of the Sine Rule. If your calculation yields an acute angle but the diagram or context suggests an obtuse angle (greater than $90^{\circ}$), you must subtract your result from $180^{\circ}$.

Find the length of side $a$ in $\triangle ABC$ where $A = 42^{\circ}$, $B = 53^{\circ}$, and $b = 5\text{ cm}$. Give your answer to 3 significant figures.

Step 1: Identify the opposite pairs. Side $a$ is opposite $42^{\circ}$, and the $5\text{ cm}$ side is opposite $53^{\circ}$.

Step 2: Substitute the pairs into the Sine Rule formula.

$$\frac{a}{\sin 42^{\circ}} = \frac{5}{\sin 53^{\circ}}$$

Step 3: Rearrange to solve for $a$ and calculate.

$$a = \frac{5}{\sin 53^{\circ}} \times \sin 42^{\circ}$$ $$a = 4.1891... \text{ cm}$$

Final Answer: $4.19\text{ cm}$

In $\triangle ABC$, $AB = 8.1\text{ cm}$, $BC = 12.3\text{ cm}$, and $\angle ACB = 27^{\circ}$. Calculate the size of $\angle BAC$ (let's call it $x$). Give your answer to 1 decimal place.

Step 1: Identify the opposite pairs. The angle $x$ is opposite $12.3\text{ cm}$, and the angle $27^{\circ}$ is opposite $8.1\text{ cm}$.

Step 2: Substitute into the flipped Sine Rule formula.

$$\frac{\sin x}{12.3} = \frac{\sin 27^{\circ}}{8.1}$$

Step 3: Rearrange to solve for $\sin x$.

$$\sin x = \frac{\sin 27^{\circ} \times 12.3}{8.1}$$ $$\sin x = 0.6894...$$

Step 4: Use the inverse sine function to find $x$.

$$x = \sin^{-1}(0.6894...)$$ $$x = 43.582...^{\circ}$$

Final Answer: $43.6^{\circ}$

The Cosine Rule

If a triangle is completely locked down by its three sides, its internal angles are mathematically guaranteed. The Cosine Rule extends Pythagoras' Theorem for non-right-angled triangles.

To find a missing side (provided on the Edexcel Higher Tier formula sheet):

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

To find a missing angle (you must rearrange the formula yourself):

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

The critical rule is that side $a$ must always be the side opposite the angle $A$ you are using or finding; sides $b$ and $c$ are entirely interchangeable. If you are calculating an angle, use the inverse cosine ($\cos^{-1}$) function. Unlike the Sine Rule, there is no ambiguous case here—if an angle is obtuse, $\cos A$ will naturally be negative, and your calculator will provide the correct value automatically.

In $\triangle ABC$, $AB = 7\text{ cm}$, $AC = 10\text{ cm}$, and $\angle BAC = 60^{\circ}$. Calculate the length of $BC$ (let's call it $a$). Give your answer to 1 decimal place.

Step 1: Substitute the values into the Cosine Rule formula for sides.

$$a^2 = 10^2 + 7^2 - 2(10)(7)\cos 60^{\circ}$$

Step 2: Calculate the terms carefully, following the order of operations.

$$a^2 = 100 + 49 - 140(0.5)$$ $$a^2 = 149 - 70 = 79$$

Step 3: Square root to find the length of $a$.

$$a = \sqrt{79}$$ $$a = 8.888... \text{ cm}$$

Final Answer: $8.9\text{ cm}$

In $\triangle ABC$, $AB = 7\text{ cm}$, $BC = 5\text{ cm}$, and $AC = 8\text{ cm}$. Find the size of $\angle ABC$ to 1 decimal place.

Step 1: Identify $a$, $b$, and $c$. We want to find angle $B$, so side $a$ in our formula must be the side opposite $B$, which is $8\text{ cm}$. Sides $b$ and $c$ are $7\text{ cm}$ and $5\text{ cm}$.

Step 2: Substitute into the rearranged Cosine Rule formula for angles.

$$\cos B = \frac{7^2 + 5^2 - 8^2}{2 \times 7 \times 5}$$

Step 3: Simplify the fraction.

$$\cos B = \frac{49 + 25 - 64}{70} = \frac{10}{70}$$

Step 4: Use the inverse cosine function.

$$B = \cos^{-1}\left(\frac{10}{70}\right)$$ $$B = 81.786...^{\circ}$$

Final Answer: $81.8^{\circ}$

Calculator Precision and Etiquette

A single premature rounding can cost you your final accuracy mark in an exam. Always check that your calculator is set to Degrees (D or DEG mode), as Radians (R) or Gradians (G) will produce entirely wrong answers.

Never round values during intermediate steps, such as writing down $\sin 42^{\circ}$ as $0.67$ midway through a calculation. Instead, use exact fractions or the "ANS" button on your calculator to carry the full decimal value to the very final step before rounding to Edexcel's standard (3 significant figures for sides, 1 decimal place for angles, unless stated otherwise).