Geometry and Trigonometry - Triangles – Sine and Cosine rules

In $\triangle ABC$, $b = 7$ cm, $c = 10$ cm, and $\angle A = 45^\circ$. Find the length of side $a$ to two decimal places.

Using the Cosine Rule: $a^2 = b^2 + c^2 - 2bc \cos A$ $a^2 = 7^2 + 10^2 - 2(7)(10) \cos 45^\circ$ $a^2 = 49 + 100 - 140 \left(\frac{\sqrt{2}}{2}\right)$ $a^2 = 149 - 70\sqrt{2}$ $a^2 \approx 149 - 98.9949 = 50.0051$ $a \approx \sqrt{50.0051} \approx 7.07 \text{ cm}$

Since we are given two sides and the included angle (SAS), we apply the Cosine Rule to find the missing opposite side.

In $\triangle PQR$, $q = 12$ cm, $\angle P = 40^\circ$, and $\angle Q = 60^\circ$. Find the length of side $p$.

Using the Sine Rule: $\frac{p}{\sin P} = \frac{q}{\sin Q}$ $\frac{p}{\sin 40^\circ} = \frac{12}{\sin 60^\circ}$ $p = \frac{12 \sin 40^\circ}{\sin 60^\circ}$ $p \approx \frac{12 \times 0.6428}{0.8660}$ $p \approx 8.91 \text{ cm}$

We use the Sine Rule because we have a known side-angle pair ($q$ and $\angle Q$) and we need to find a side corresponding to another known angle ($\angle P$).

Find the area of a triangle with sides $8$ cm and $5$ cm and an included angle of $30^\circ$.

Using the Area formula: $\text{Area} = \frac{1}{2}ab \sin C$ $\text{Area} = \frac{1}{2} (8)(5) \sin 30^\circ$ $\text{Area} = 20 \times 0.5 = 10 \text{ cm}^2$

The area of a triangle is half the product of two sides and the sine of the angle between them.

1. Identify the largest angle: The largest angle is opposite the longest side, $XY = 15$ cm. Let this be angle $Z$.
2. Apply the Cosine Rule: $\cos Z = \frac{10^2 + 12^2 - 15^2}{2 \times 10 \times 12}$
3. Calculate the value: $\cos Z = \frac{100 + 144 - 225}{240} = \frac{19}{240} \approx 0.07917$
4. Find the angle: $Z = \cos^{-1}(0.07917) \approx 85.46^\circ$
5. Rounding to the nearest degree gives $85^\circ$.

To find an angle when all three sides are known, the Cosine Rule is required. Since we want the largest angle, we target the one opposite the side with length 15.

1. Find the third angle $C$: $C = 180^\circ - (35^\circ + 75^\circ) = 70^\circ$
2. Use the Sine Rule to find side $a$: $\frac{a}{\sin 35^\circ} = \frac{8}{\sin 70^\circ}$ $a = \frac{8 \sin 35^\circ}{\sin 70^\circ} \approx \frac{8 \times 0.5736}{0.9397} \approx 4.883 \text{ cm}$
3. Calculate the area using sides $a, c$ and included angle $B$: $\text{Area} = \frac{1}{2} \times a \times c \times \sin B$ $\text{Area} = \frac{1}{2} \times 4.883 \times 8 \times \sin 75^\circ$ $\text{Area} \approx 19.532 \times 0.9659 \approx 18.9 \text{ cm}^2$

To find the area, we need two sides and their included angle. We first find the missing angle to use the Sine Rule for a second side, then apply the sine area formula.