Trigonometry I
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Master the fundamental trigonometric ratios (Sine, Cosine, and Tangent) to calculate unknown sides and acute angles in any right-angled triangle.
1. The Anatomy of a Right-Angled Triangle
In any right-angled triangle, side names are always defined relative to a chosen reference angle \(\theta\) (where \(\theta \neq 90^\circ\)):
- Hypotenuse (H): The longest side, always directly opposite the \(90^\circ\) right angle.
- Opposite (O): The side directly across from the reference angle \(\theta\).
- Adjacent (A): The side next to the reference angle \(\theta\) that connects it to the right angle.
Interactive Ratio Visualizer
Adjust the angle \(\theta\) to watch how the side lengths and ratios change in real-time:
0.57
0.82
0.70
2. The Invariance of Trigonometric Ratios
Why do these ratios matter? Because for any two right triangles with the same acute angle \(\theta\), the triangles are similar. Regardless of whether the triangle is as small as an exercise book or as large as Mount Kenya, the ratio of side lengths depends solely on the angle \(\theta\).
Key Formulas
Primary Trigonometric Ratios (SOH CAH TOA)
\[\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad (\text{SOH})\]\[\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad (\text{CAH})\]\[\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \quad (\text{TOA})\]Inverse Ratios (Finding Unknown Angles)
When side lengths are known and the angle \(\theta\) is unknown, apply the inverse trigonometric functions:
\[\theta = \sin^{-1}\left(\frac{\text{Opp}}{\text{Hyp}}\right) \qquad \theta = \cos^{-1}\left(\frac{\text{Adj}}{\text{Hyp}}\right) \qquad \theta = \tan^{-1}\left(\frac{\text{Opp}}{\text{Adj}}\right)\]Angles of Elevation and Depression
- Angle of Elevation: The angle measured upwards from the horizontal line of sight to an object.
- Angle of Depression: The angle measured downwards from the horizontal line of sight to an object.
- Crucial Geometric Property: By alternate interior angles between parallel horizontals, \(\text{Angle of Elevation} = \text{Angle of Depression}\).
Worked Examples
Level 1: Finding an Unknown Side (Easy)
Problem: An electrical technician leans a \(12\text{ m}\) ladder against a wall at an angle of \(30^\circ\) to the horizontal ground. Calculate the vertical height the ladder reaches up the wall.
- Identify Given & Required: Hypotenuse \(H = 12\text{ m}\), Angle \(\theta = 30^\circ\), Opposite side \(O = h\) (height).
- Select the Ratio: \(\sin\theta = \frac{O}{H}\) connects Opposite and Hypotenuse (SOH).
- Substitute Values: \[\sin(30^\circ) = \frac{h}{12}\]
- Solve for \(h\): Since \(\sin(30^\circ) = 0.5\), \[h = 12 \times 0.5 = 6\text{ metres}.\]
Level 2: Finding an Unknown Angle (Medium)
Problem: A ramp is built to provide wheelchair access to a clinic in Nakuru. The ramp rises \(1.5\text{ m}\) vertically over a horizontal distance of \(8.5\text{ m}\). Calculate the angle of inclination of the ramp to one decimal place.
- Identify Given & Required: Opposite \(O = 1.5\text{ m}\), Adjacent \(A = 8.5\text{ m}\), Angle \(\theta = ?\)
- Select the Ratio: \(\tan\theta = \frac{O}{A}\) relates Opposite and Adjacent (TOA).
- Set up and apply Inverse Tangent: \[\tan\theta = \frac{1.5}{8.5} = 0.17647\]\[\theta = \tan^{-1}(0.17647) \approx 10.0^\circ.\]
Level 3: Multi-Step Real World Application (Hard)
Problem: A surveyor stands \(40\text{ m}\) away from the base of a telecommunications mast. Using a theodolite on a tripod \(1.6\text{ m}\) high, she measures the angle of elevation to the top of the mast as \(38^\circ\). What is the total height of the mast to one decimal place?
- Define Triangle Geometry: The triangle sits above the tripod height. Base (Adjacent) \(= 40\text{ m}\), angle \(\theta = 38^\circ\), and upper portion of mast is Opposite (\(y\)).
- Find \(y\) using Tangent: \[\tan(38^\circ) = \frac{y}{40} \implies y = 40 \times \tan(38^\circ)\]\[y = 40 \times 0.78129 = 31.25\text{ m}.\]
- Add Tripod Height: \[\text{Total Height} = y + 1.6 = 31.25 + 1.6 = 32.85 \approx 32.9\text{ m}.\]
Common Mistakes
Misconception 1: Mixing up Opposite and Adjacent when Reference Angle Changes
The Mistake: Assuming the bottom side is always 'Adjacent' and the vertical side is always 'Opposite'.
The Correction: 'Opposite' and 'Adjacent' are defined relative to the specific angle you are considering. If you look from the top vertex angle, the horizontal base becomes the Opposite side, and the vertical upright becomes the Adjacent side.
Misconception 2: Misinterpreting the Angle of Depression
The Mistake: Drawing the angle of depression between the line of sight and the vertical wall or tower.
The Correction: The angle of depression is ALWAYS measured downwards from a horizontal reference line, never from the vertical.
Misconception 3: Calculator in Radian Mode instead of Degree Mode
The Mistake: Evaluating \(\sin(30^\circ)\) and getting \(-0.988\) instead of \(0.5\).
The Correction: Ensure your scientific calculator is in DEG (Degree) mode when working with geometric angles given in degrees.
Real World
Practice