Sine and Cosine Rules

Extending trigonometry beyond right-angled triangles

The basic sine, cosine and tangent ratios apply directly to right-angled triangles. The sine rule and cosine rule allow us to solve any triangle, including one with no right angle. To solve a triangle means to find one or more of its unknown side lengths or angles.

Triangle notation is essential. Vertices are labelled $A$, $B$ and $C$, while the side opposite each vertex uses the matching lower-case letter: $a$ is opposite $A$, $b$ is opposite $B$, and $c$ is opposite $C$.

Choosing the correct rule

Choose a rule from the information given rather than from the appearance of the diagram.

Use the sine rule when you know an opposite side–angle pair and also know, or need to find, another side or its opposite angle. This commonly gives ASA, AAS or SSA information.

Use the cosine rule when:

• two sides and their included angle are known and the third side is required (SAS); or
• all three sides are known and an angle is required (SSS).

A useful first step is to mark the known values on a sketch and identify every opposite side–angle pair.

Using the sine rule to find a side

The sine rule is

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

Only two fractions are needed in any one calculation. If $a=8\text{ cm}$, $A=42^\circ$ and $B=71^\circ$, then $a$ and $A$ form the known opposite pair. To find $b$:

$$\frac{8}{\sin42^\circ}=\frac{b}{\sin71^\circ}$$ $$b=\frac{8\sin71^\circ}{\sin42^\circ}\approx11.3\text{ cm}.$$

The answer is reasonable because $B>A$, so the opposite side $b$ should be longer than $a$.

Using the sine rule to find an angle

When the unknown is an angle, it is often convenient to invert the sine rule:

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

For example, suppose $a=7\text{ cm}$, $A=38^\circ$ and $b=9\text{ cm}$. Then

$$\frac{\sin B}{9}=\frac{\sin38^\circ}{7},$$

so

$$\sin B=\frac{9\sin38^\circ}{7}.$$

Using inverse sine gives $B\approx52.3^\circ$. Make sure the calculator is in degree mode.

There may also be a second answer. Since

$$\sin\theta=\sin(180^\circ-\theta),$$

the supplementary angle $180^\circ-52.3^\circ=127.7^\circ$ has the same sine. Here both angles are possible because $38^\circ+127.7^\circ<180^\circ$. In another question, the supplementary angle may make the angle total at least $180^\circ$, in which case it must be rejected. A diagram or further information may also rule it out.

Using the cosine rule to find a side

The cosine rule can be written as

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

The side isolated on the left, $a$, must be opposite angle $A$. The other two sides form that angle.

Suppose $b=7\text{ cm}$, $c=10\text{ cm}$ and the included angle $A=48^\circ$. Then

$$a^2=7^2+10^2-2(7)(10)\cos48^\circ,$$

so

$$a=\sqrt{7^2+10^2-2(7)(10)\cos48^\circ}\approx7.4\text{ cm}.$$

The positive square root is used because a length is positive. When $A=90^\circ$, $\cos90^\circ=0$, so the final term disappears and the formula becomes $a^2=b^2+c^2$: Pythagoras’ theorem is therefore a special case of the cosine rule.

Using the cosine rule to find an angle

Rearranging the same formula gives

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

If $a=9\text{ cm}$, $b=7\text{ cm}$ and $c=12\text{ cm}$, then

$$A=\cos^{-1}\left(\frac{7^2+12^2-9^2}{2(7)(12)}\right)\approx48.2^\circ.$$

The subtraction order matters: the square of the side opposite the required angle is subtracted in the numerator. Equivalent rearrangements give

$$\cos B=\frac{a^2+c^2-b^2}{2ac} \quad\text{and}\quad \cos C=\frac{a^2+b^2-c^2}{2ab}.$$

Completing and checking a triangle

After finding one value, the remaining values can sometimes be obtained using the same rule, the other rule, or the angle sum $A+B+C=180^\circ$. Choose the route that uses the information most directly.

Check the completed triangle geometrically: its angles must be positive and total $180^\circ$; the longest side must lie opposite the largest angle; and any two sides must have a combined length greater than the third side. These checks can reveal a mismatched side and angle, an incorrect rearrangement or premature rounding.

Match sides and angles

Side $a$ is opposite $A$, $b$ is opposite $B$, and $c$ is opposite $C$.

Sine rule

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

Use it when a known opposite side–angle pair is available. For an unknown angle, the inverted form may be easier:

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

When using inverse sine, test the supplementary angle $180^\circ-\theta$ if the given information could form two triangles.

Cosine rule

For a missing side with SAS information:

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

For a missing angle with SSS information:

$$A=\cos^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right).$$

The side on the left is always opposite the named angle.

Final checks

Use degree mode, retain unrounded calculator values, and round only at the end. Check that angles total $180^\circ$ and that the largest side is opposite the largest angle.