Grade 10 core mathematics Measurements and Geometry – Trigonometry I Notes
2.4 Trigonometry I
Topic: 2.0 Measurements and Geometry — Subject: Core Mathematics — Target age: 15 (Kenya)
- (a) Identify and outline sub-sub-strands:
- Trigonometric ratios of acute angles
- Sines and cosines of complementary angles
- Trigonometric ratios of special angles
- Application of trigonometric ratios
- (b) Determine the trigonometric ratios of acute angles from mathematical tables and calculators.
- (c) Relate sines and cosines of complementary angles (sin θ = cos(90° − θ)).
- (d) Relate the sine, cosine and tangent ratios of acute angles (tan θ = sin θ / cos θ).
- (e) Determine trigonometric ratios of special angles 30°, 45°, 60°, 90° using triangles.
- (f) Apply trigonometric ratios to angles of elevation and depression.
- (g) Reflect on and describe uses of trigonometry in real life (surveying, construction, navigation, etc.).
Key definitions and ratios
In a right-angled triangle, for an acute angle θ:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent = sin θ / cos θ
Mnemonic (useful): SOH-CAH-TOA
Right triangle (visual)
Label sides relative to angle θ: opposite, adjacent, hypotenuse.
Complementary angles and identities
Complementary angles: If two angles add to 90°, then they are complementary.
Relation: sin θ = cos(90° − θ) and cos θ = sin(90° − θ).
Pythagorean identity: sin²θ + cos²θ = 1.
Special angles: 30°, 45°, 60°, 90° (derivations)
Take legs = 1, 1 → hypotenuse = √2.
- sin 45° = 1 / √2 = √2 / 2 ≈ 0.707
- cos 45° = 1 / √2 = √2 / 2 ≈ 0.707
- tan 45° = 1
Take equilateral triangle side 2, drop altitude → two right triangles with sides 1, √3, 2.
- sin 30° = 1 / 2 = 0.5
- cos 30° = √3 / 2 ≈ 0.866
- tan 30° = 1 / √3 ≈ 0.577
- sin 60° = √3 / 2 ≈ 0.866
- cos 60° = 1 / 2 = 0.5
- tan 60° = √3 ≈ 1.732
90°: sin 90° = 1, cos 90° = 0, tan 90° is undefined (adjacent side = 0).
Using mathematical tables and calculators
Calculator (common steps):
- Ensure calculator is in Degree mode (not Radians).
- To find sin 37°: press sin → 37 → = → read value (≈ 0.6018).
- To find an angle from ratio: use inverse function, e.g. angle = sin⁻¹(0.5) = 30°.
Example table snippet (for classroom use):
| Angle (°) | sin | cos |
|---|---|---|
| 30 | 0.5000 | 0.8660 |
| 45 | 0.7071 | 0.7071 |
| 60 | 0.8660 | 0.5000 |
(Use table values for quick look-up; calculators give more digits.)
Worked examples
From a point 20 m from the base of a tree the angle of elevation to the top is 30°. Find the tree height (to 2 d.p.).
tan 30° = opposite / adjacent = height / 20
height = 20 × tan 30° = 20 × (1/√3) ≈ 20 × 0.57735 = 11.55 m
Find θ if sin θ = 0.6. Use calculator.
θ = sin⁻¹(0.6) ≈ 36.87° (round to 2 d.p. if needed).
Angles of elevation and depression
Definition:
- Angle of elevation — angle from horizontal up to an object (e.g., top of tower).
- Angle of depression — angle from horizontal down to an object (e.g., base of a hill) — equals the corresponding angle of elevation from the object to the observer.
Simple classroom activity: measure angle of elevation to a flagpole using a protractor-clinometer (or smartphone app). Measure horizontal distance, use tan to compute height.
Suggested learning experiences (suitable for Kenyan schools, age 15)
- Start with concrete triangles: draw and measure sides, identify opposite/adjacent/hypotenuse for given angle.
- Use paper folding to make 45° and 30° triangles (isosceles right & equilateral split) to derive special-angle ratios.
- Calculator practice: switch between Degrees and Radians, find values and inverse values, graph short worksheet.
- Table lookup exercise: give students a small trigonometric table and ask them to find values and interpolate if needed.
- Practical outdoors: measure the height of a tree, flagpole or wall using a clinometer or smartphone, a tape measure and trigonometry. Work in pairs/groups.
- Problem-solving: set varied real-life problems (ladder leaning against a wall, slope of a road, height of a mast using angle of elevation).
- Group reflection: discuss careers and fields that use trigonometry (surveying, civil engineering, architecture, navigation, meteorology, ICT and animation).
Practice exercises (for class or homework)
- Find sin, cos and tan of 30°, 45°, 60° (exact values and 3 decimal approximations).
- Using a calculator, find sin 37°, cos 74°, tan 23° (3 d.p.).
- A ladder 5 m long leans against a wall making an angle of 70° with the ground. How high up the wall does it reach? (3 s.f.)
- From a point 50 m from a tower the angle of elevation is 25°. Find the tower height.
- Explain in a short paragraph one way trigonometry is useful in farming, building or surveying in Kenya.
Assessment and resources
Assessment: short quiz on definitions and special angle values, calculator test, practical measurement task and written word problems.
Resources: scientific calculators, trigonometric tables, protractors/clinometers, tape measures, prepared triangle sheets, smartphones with clinometer apps, classroom whiteboard.