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Learning Resources

Trigonometry

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 09 Pathway: N/A

First Principles

Objective: Master the basic trigonometric ratios (Sine, Cosine, Tangent) to solve for unknown side lengths and angles in right-angled triangles.

Interactive SOH CAH TOA Explorer

Adjust the angle (\(\theta\)) or hypotenuse length to see how opposite and adjacent sides scale!

Adj (A) Opp (O) Hyp (H) θ
sin(\(\theta\)) = O/H
0.574
cos(\(\theta\)) = A/H
0.819
tan(\(\theta\)) = O/A
0.700

(a) Concrete Scenario: Imagine positioning a ladder against a wall or building a wheelchair ramp in Nakuru. The angle of elevation \(\theta\) determines how steep the incline is. Trigonometry lets us calculate how high the ladder reaches or how long the ramp must be without physically measuring them on site.

(b) Defining the Sides: In a right-angled triangle relative to acute angle \(\theta\):

  • Hypotenuse (H): Longest side, always directly opposite the \(90^\circ\) right angle.
  • Opposite (O): The leg directly across from angle \(\theta\).
  • Adjacent (A): The leg alongside angle \(\theta\) (excluding the hypotenuse).

(c) SOH CAH TOA Memory Tool: \[\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad (\text{SOH})\] \[\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad (\text{CAH})\] \[\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \quad (\text{TOA})\]

Pro-Tip: Always identify the Hypotenuse first, then label Opposite and Adjacent relative to the angle you are working with!

Key Formulas

\(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\implies \text{Opposite} = \text{Hypotenuse} \times \sin\theta\)
\(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\implies \text{Adjacent} = \text{Hypotenuse} \times \cos\theta\)
\(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\implies \text{Opposite} = \text{Adjacent} \times \tan\theta\)
\(\theta = \sin^{-1}\left(\frac{\text{Opp}}{\text{Hyp}}\right) = \cos^{-1}\left(\frac{\text{Adj}}{\text{Hyp}}\right) = \tan^{-1}\left(\frac{\text{Opp}}{\text{Adj}}\right)\) — Inverse trigonometric ratios used to find angles.

Worked Examples

Example 1 (Easy - Calculating Crane Arm Length): A construction crane in Nairobi lifts a beam to a wall height of \(75\text{ m}\). The arm makes an angle of \(55^\circ\) with the ground. Find the length of the crane arm (hypotenuse) to 1 decimal place.
  1. Identify sides relative to \(55^\circ\): \(\text{Opposite} = 75\text{ m}\), \(\text{Hypotenuse} = h\).
  2. Choose ratio: Sine uses Opp and Hyp (\(\text{SOH}\)).
  3. Set up equation: \(\sin(55^\circ) = \frac{75}{h}\).
  4. Rearrange: \(h = \frac{75}{\sin(55^\circ)} = \frac{75}{0.8192} \approx 91.6\text{ m}\).
  5. Answer: \(91.6\text{ m}\).
Example 2 (Medium - Angle of Elevation): A matatu ramp rises \(5.4\text{ m}\) vertically over a sloped surface length of \(27\text{ m}\). Find the angle of elevation \(\theta\) to 1 decimal place.
  1. Identify sides: \(\text{Opposite} = 5.4\text{ m}\), \(\text{Hypotenuse} = 27\text{ m}\).
  2. Choose ratio: \(\sin\theta = \frac{\text{Opp}}{\text{Hyp}} = \frac{5.4}{27} = 0.2\).
  3. Apply inverse sine: \(\theta = \sin^{-1}(0.2) \approx 11.53^\circ\).
  4. Answer: \(11.5^\circ\).
Example 3 (Hard - Drone Angle of Depression): A drone hovering \(80\text{ m}\) vertically above a tea farm in Kericho looks down at a target patch with an angle of depression of \(40^\circ\). Calculate the horizontal distance from the drone to the target patch.
  1. By alternate interior angles, the angle of elevation from the ground target to the drone is also \(40^\circ\).
  2. Identify sides: \(\text{Opposite} = 80\text{ m}\), \(\text{Adjacent} = d\).
  3. Choose ratio: \(\tan(40^\circ) = \frac{\text{Opp}}{\text{Adj}} = \frac{80}{d}\).
  4. Rearrange: \(d = \frac{80}{\tan(40^\circ)} = \frac{80}{0.8391} \approx 95.34\text{ m}\).
  5. Answer: \(95.3\text{ m}\) (or \(95.4\text{ m}\) using full precision).

Common Mistakes

Mistake Mixing up Opposite and Adjacent sides when the position of the angle changes.
Correction The Opposite side is ALWAYS directly across from the angle you are analyzing. If you shift your focus to the top angle of the right triangle, the Opposite and Adjacent labels swap positions!
Mistake Having your scientific calculator in Radian (RAD) mode instead of Degree (DEG) mode when computing trig values.
Correction Always verify that your calculator screen displays 'D' or 'DEG' before solving school trigonometry problems!

Real World

Civil & Structural Engineering: Road contractors in Kenya calculate maximum safe gradients for mountain passes (e.g., the Rift Valley escarpment) using \(\tan\theta\).
Cellular Infrastructure: Telecom technicians calculate the exact length of support guy-wires needed to anchor mobile transmission towers in Nakuru.
Aviation & Navigational Radar: Air traffic controllers at JKA Airport determine aircraft glide path descent angles using sine and cosine ratios.

Practice

A construction crane in Nairobi lifts a beam to the top of a 75 m high wall. The crane's arm makes a 55 degrees angle with the horizontal ground. What is the length of the crane's arm in metres to one decimal place? (Type only the number, e.g., 91.6)
Review the concepts above.
A ramp at an overpass rises 5.4 m vertically over a total slope length of 27 m. What is the angle of elevation of the ramp in degrees? (Give your answer to 1 decimal place) (Type only the number, e.g., 11.5)
Review the concepts above.
A guy-wire is attached to the top of a 20 m tall grain silo in Trans Nzoia and anchored to the ground, forming a 70 degrees angle with the ground. What is the length of the wire in metres to 1 decimal place? (Type only the number, e.g., 21.3)
Review the concepts above.
A drone operates 80 m vertically above tea fields in Kericho. The camera measures a 40 degrees angle of depression to a field target. What is the horizontal distance from the drone to the target in metres to 1 decimal place? (Type only the number, e.g., 95.3)
Review the concepts above.
A phone tower technician works at a height of 38 m above the ground in Nakuru. He looks down at an equipment case at an angle of depression of 52 degrees. How far horizontally is the case from the base of the tower in metres to 1 decimal place? (Type only the number, e.g., 29.7)
Review the concepts above.
A farmer inspects crops with a drone hovering 120 m vertically above her shamba. The camera is angled downward at a 55 degrees angle of depression. What is the direct straight-line distance from the drone to the crop patch in metres to 1 decimal place? (Type only the number, e.g., 146.5)
Review the concepts above.