Grade 10 essential mathematics Measurements and Geometry – Trigonometry Notes

2.3 Trigonometry

Topic: 2.0 Measurements and Geometry — Subject: Essential Mathematics — Target age: 15 (Kenya)

Specific learning outcomes

• (a) Identify and outline the sub-sub-strands:
  • Sines and cosines of complementary angles
  • Application of trigonometric ratios (angles of elevation and depression)
• (b) Identify the sides of a right-angled triangle in relation to a particular angle
• (c) Determine the sine, cosine and tangent of acute angles in a right-angled triangle
• (d) Relate sines and cosines of complementary angles
• (e) Apply trigonometric ratios to angles of elevation and depression
• (f) Recognise the use of trigonometry in real-life situations

Quick definitions

Right-angled triangle: one angle is 90°. For a chosen acute angle θ:

• Hypotenuse — side opposite the right angle (longest side).
• Opposite — side opposite the chosen angle θ.
• Adjacent — side next to θ (not the hypotenuse).

Basic trigonometric ratios

For acute angle θ in a right triangle:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent = sin θ / cos θ

Worked examples

Sines and cosines of complementary angles

If two acute angles are complementary, their sum is 90°. For any acute θ:

sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

Reason: in the same right triangle the opposite side for θ is the adjacent side for 90° − θ, and vice versa.

Angles of elevation and depression

• Angle of elevation: angle from horizontal upward to an object (observer on ground looking up).
• Angle of depression: angle from horizontal downward to an object (observer at height looking down).
• In practical problems, both angles form right triangles; use sin, cos or tan depending on which sides are known.

Typical exam-style problems (practice)

1. A ladder 10 m long leans against a wall making an angle of 70° with the ground. How high up the wall does the ladder reach? (Give answer to 1 decimal place.)
2. From a point on the ground a man observes the top of a flagpole at an angle of elevation of 45°. If he is 20 m from the foot of the pole, find the height of the pole.
3. A radio mast casts a shadow 120 m long when the sun's rays make an angle of elevation of 35°. Find the mast height (to the nearest metre).
4. Give numerical values: sin 30°, cos 60°, tan 45°.

Answers to practice

1. height = 10 × sin 70° ≈ 10 × 0.9397 ≈ 9.4 m.
2. angle 45° → tan 45° = 1 = height / 20 → height = 20 m.
3. height = 120 × tan 35° ≈ 120 × 0.7002 ≈ 84 m (nearest metre).
4. sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1.

Real-life uses (Kenyan context examples)

• Surveying land and measuring heights of trees, towers, or buildings using angles and distances (useful in agriculture, forestry, town planning).
• Designing roofs and determining slopes (construction and carpentry).
• Navigation and bearings for road or bridge works; locating positions using angles.
• Estimating heights of mobile-phone masts or transmission towers from a safe distance.

Suggested learning experiences

• Practical measuring: use a protractor or simple clinometer, tape measure and record distances and angles to estimate heights (e.g., school flagpole, mango tree).
• Group activity: measure shadows at a fixed time, use angle of elevation of the sun or shadow method to estimate tree height and compare with a direct measure.
• Classroom sketching: draw right triangles, label sides for different θ, and practise calculating ratios without a calculator for special angles 30°, 45°, 60°.
• Problem-solving: set real-life problems (roof slope, ladder safety angle, height of a building) and present solutions using trigonometry.
• Use calculators in degree mode and check results manually where possible (e.g., known exact ratios for 30°, 45°, 60°).

Teaching tips & assessment

• Start from concrete (measurements on school grounds) before moving to abstract ratios and formulae.
• Assess by giving short practical tasks (measure and compute) and written exercises (identifying opposite/adjacent/hypotenuse and computing ratios).
• Encourage use of units (metres) and correct rounding. Remind learners to set calculators to degrees.

End of notes for 2.3 Trigonometry. These notes give the essentials for age 15 learners in Kenya: definitions, diagrams, worked examples, practice and classroom activities to meet the specific learning outcomes (a)–(f).