Sine/Cosine Rules & Graphs
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Core Objective: Master the Sine and Cosine rules for solving non-right-angled triangles, and understand transformations of trigonometric wave graphs.
The Surveyor's Dilemma: A land surveyor working along the banks of the Tana River in Garissa needs to find the exact distance across the river between two boundary beacons, \(B\) and \(C\). Since wading across the swollen river is impossible, she sets up a baseline along her side from \(A\) to \(B\) measuring \(120\text{ m}\), and measures \(\angle CAB = 65^\circ\) and \(\angle ABC = 75^\circ\). Because \(\triangle ABC\) contains no right angle, standard basic ratios (\(\sin = \frac{O}{H}\)) cannot solve this directly. She needs universal laws: the Sine Rule and Cosine Rule.
1. The Sine Rule (Law of Sines):
In any triangle \(ABC\), dropping a perpendicular height \(h\) from vertex \(C\) to side \(c\) shows that \(h = b\sin A\) and \(h = a\sin B\). Equating these gives:
\[a\sin B = b\sin A \implies \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\]where \(R\) is the circumradius of the triangle. The Sine Rule is used when we know a side and its opposite angle pair (AAS or SSA).
2. The Cosine Rule (Law of Cosines):
When we know two sides and the included angle (SAS) or all three sides (SSS), we cannot immediately form an opposite pair. Applying Pythagoras' Theorem on the split triangle yields the generalised Pythagoras theorem:
\[c^2 = a^2 + b^2 - 2ab\cos C\]Notice that if \(C = 90^\circ\), \(\cos 90^\circ = 0\), returning directly to \(c^2 = a^2 + b^2\).
3. Trigonometric Waves:
As a point rotates counter-clockwise around a unit circle at an angle \(\theta\), its coordinates are \((\cos\theta, \sin\theta)\). Plotting height over angle produces sinusoidal waves of the general form \(y = A\sin(Bx + C) + D\), where \(|A|\) is the amplitude, \(T = \frac{360^\circ}{|B|}\) is the period, and \(D\) is the vertical shift.
Key Formulas
1. The Sine Rule
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]Use when given: (Two angles and one side) or (Two sides and a non-included angle).
2. The Cosine Rule (Solving for Side)
\[a^2 = b^2 + c^2 - 2bc\cos A\] \[b^2 = a^2 + c^2 - 2ac\cos B\] \[c^2 = a^2 + b^2 - 2ab\cos C\]Use when given two sides and the included angle (SAS).
3. The Cosine Rule (Solving for Angle)
\[\cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}\]Use when all three sides are known (SSS).
4. Area of Any Triangle
\[\text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B\]5. Trigonometric Wave Characteristics
For the wave equation \(y = A\sin(Bx + C) + D\) or \(y = A\cos(Bx + C) + D\):
- Amplitude: \(|A| = \frac{\text{Maximum} - \text{Minimum}}{2}\)
- Period (\(T\)): \(T = \frac{360^\circ}{|B|}\) (or \(\frac{2\pi}{|B|}\) radians)
- Phase Shift: \(-\frac{C}{B}\)
- Vertical Shift / Mean Line: \(y = D\)
Worked Examples
In \(\triangle ABC\), \(\angle A = 30^\circ\), \(\angle B = 45^\circ\), and side \(a = 10\text{ cm}\). Find the length of side \(b\) correct to 1 decimal place.
- Identify the appropriate rule: We are given two angles (\(A, B\)) and one corresponding opposite side (\(a\)). Apply the Sine Rule: \[\frac{a}{\sin A} = \frac{b}{\sin B}\]
- Substitute the known values: \[\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}\]
- Evaluate trigonometric values: Recall \(\sin 30^\circ = 0.5\) and \(\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071\). \[\frac{10}{0.5} = \frac{b}{0.7071} \implies 20 = \frac{b}{0.7071}\]
- Solve for \(b\): \[b = 20 \times 0.7071 = 14.142\text{ cm} \approx 14.1\text{ cm}\]
A triangular plot of land in Nakuru has boundaries measuring \(a = 7\text{ m}\), \(b = 8\text{ m}\), and \(c = 9\text{ m}\). Calculate the size of angle \(C\) opposite the longest boundary, rounded to the nearest whole degree.
- Identify the rule: All three sides are known (SSS). Use the Cosine Rule rearranged for \(\cos C\): \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]
- Substitute the side lengths: \[\cos C = \frac{7^2 + 8^2 - 9^2}{2(7)(8)} = \frac{49 + 64 - 81}{112}\]
- Simplify the numerator and denominator: \[\cos C = \frac{113 - 81}{112} = \frac{32}{112} = \frac{2}{7} \approx 0.2857\]
- Compute the inverse cosine: \[C = \cos^{-1}(0.2857) = 73.398^\circ \approx 73^\circ\]
Two fishing dhows leave a port in Lamu at the same time. Dhow \(P\) sails on a bearing of \(040^\circ\) at \(12\text{ km/h}\), while Dhow \(Q\) sails on a bearing of \(100^\circ\) at \(15\text{ km/h}\). Calculate the distance between the two dhows after \(2\text{ hours}\), rounded to 1 decimal place.
- Find the distances travelled:
- Distance of \(P\) from port: \(d_P = 12\text{ km/h} \times 2\text{ h} = 24\text{ km}\)
- Distance of \(Q\) from port: \(d_Q = 15\text{ km/h} \times 2\text{ h} = 30\text{ km}\)
- Find the included angle \(\theta\): The angle between their paths is the difference in bearings: \[\theta = 100^\circ - 040^\circ = 60^\circ\]
- Apply the Cosine Rule to find distance \(d = PQ\): \[d^2 = d_P^2 + d_Q^2 - 2(d_P)(d_Q)\cos(60^\circ)\]
- Substitute values: \[d^2 = 24^2 + 30^2 - 2(24)(30)\cos(60^\circ)\] \[d^2 = 576 + 900 - 1440(0.5) = 1476 - 720 = 756\]
- Calculate square root: \[d = \sqrt{756} \approx 27.495\text{ km} \approx 27.5\text{ km}\]
Common Mistakes
Real World
Real-World Pan-African Applications of Advanced Trigonometry:
1. SGR Civil Engineering & Triangulation across the Great Rift Valley:
During the construction of the Standard Gauge Railway (SGR) Phase 2A through the escarpments of the Great Rift Valley in Mai Mahiu, engineers faced jagged hills and deep gorges. Direct tape measurement was impossible. By establishing fixed reference towers and measuring angles with optical theodolites, surveyors used the Sine and Cosine Rules (Triangulation) to compute tunnel alignments through the Ngong Hills with sub-centimetre precision.
2. Solar Energy Harvesting in Northern Kenya:
At the Lake Turkana Wind and Solar installations in Marsabit, solar tracking panels adjust their tilt angle \(\theta\) throughout the day. The energy absorption rate follows the sinusoidal wave model \(P(t) = P_{\max} \sin\left(\frac{\pi t}{12}\right)\), where trigonometric transformations determine the optimal angle to capture maximum irradiance.
3. Maritime Navigation along the Swahili Coast:
Cargo vessels docking at Kilindini Harbour in Mombasa calculate clearance against tidal waves. Ocean tide levels follow sinusoidal functions: \(h(t) = A\cos(Bt) + D\). Understanding amplitude and period enables harbour masters to predict safe docking depths during spring and neap tides.
Practice