Grade 10 Mathematics · Strand 2.0 Measurements and Geometry · Sub-strand 2.4 Trigonometry 1
These are free study notes for Grade 10 learners, written to the KICD curriculum design and taken from our full Grade 10 Mathematics notes. Trigonometry turns up across Senior School Mathematics, Physics and Geography. Learn to name the sides correctly first; almost every mistake starts there.
How tall is the school flagpole? How steep is a roof or a ramp? Trigonometry answers such questions without climbing anything. It links the sides and angles of right-angled triangles through three ratios: sine, cosine and tangent. In this topic we use these ratios, the special angles 30°, 45° and 60°, and angles of elevation and depression.
Naming the sides
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Opposite side
The side facing the chosen acute angle.
Adjacent side
The side next to the chosen acute angle that is not the hypotenuse.
Watch out
The opposite and adjacent sides swap when you choose the other acute angle. The hypotenuse never changes.
The three trigonometric ratios
| Ratio | Definition | In the 3, 4, 5 triangle (angle opposite 4) |
|---|---|---|
| sin A | opposite ÷ hypotenuse | 4/5 = 0.8 |
| cos A | adjacent ÷ hypotenuse | 3/5 = 0.6 |
| tan A | opposite ÷ adjacent | 4/3 = 1.333 |
The ratios depend only on the angle, not on the size of the triangle, because triangles with the same angles are similar.
- For an acute angle, sin A and cos A are less than 1 (the hypotenuse is the longest side); tan A can be any positive number.
- As the angle increases, sin and tan increase and cos decreases.
- A calculator must be in degree mode (DEG). Use sin^-1, cos^-1 and tan^-1 to find an angle from a ratio.
Finding an angle
A ramp 6 m long rises 1.2 m.
sin A = 1.2 ÷ 6 = 0.2.
A = sin^-1 0.2 = 11.5°.
Finding sides of right-angled triangles
Method
- Sketch the triangle and mark the angle.
- Label the side you know and the side you want (O, A or H).
- Choose the ratio that uses those two sides.
- Write the equation and solve it.
A ladder against a wall
A ladder 5 m long makes 60° with the ground.
Height reached = 5 × sin 60° = 5 × 0.8660 = 4.33 m.
Distance from the wall = 5 × cos 60° = 5 × 0.5 = 2.5 m.
Check with Pythagoras: 4.33² + 2.5² = 18.75 + 6.25 = 25 = 5².
Finding the hypotenuse
The side adjacent to 40° is 12 cm. Find the hypotenuse h.
cos 40° = 12 ÷ h, so h = 12 ÷ 0.7660 = 15.7 cm.
Relationships between the ratios
| Relationship | Example |
|---|---|
| sin A = cos (90° - A) | sin 25° = cos 65° = 0.4226 |
| cos A = sin (90° - A) | cos 20° = sin 70° = 0.9397 |
| tan A = sin A ÷ cos A | tan 40° = 0.6428 ÷ 0.7660 = 0.839 |
| sin² A + cos² A = 1 | (3/5)² + (4/5)² = 9/25 + 16/25 = 1 |
Complementary angles
Solve cos 2x = sin (x + 30°).
The angles are complementary: 2x + x + 30° = 90°.
3x = 60°, so x = 20°.
From one ratio to the others
sin A = 5/13. Then cos² A = 1 - 25/169 = 144/169.
cos A = 12/13 and tan A = 5/12.
The special angles 30°, 45° and 60°
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | sqrt(3)/2 = 0.8660 | 1/sqrt(3) = 0.5774 |
| 45° | sqrt(2)/2 = 0.7071 | sqrt(2)/2 = 0.7071 | 1 |
| 60° | sqrt(3)/2 = 0.8660 | 1/2 | sqrt(3) = 1.7321 |
| 90° | 1 | 0 | Undefined |
The 45° values come from an isosceles right-angled triangle with equal sides 1 and hypotenuse sqrt(2).
Using exact values
Angle 30°, hypotenuse 12 cm: opposite = 12 × 1/2 = 6 cm.
Adjacent = 12 × sqrt(3)/2 = 6 sqrt(3), about 10.39 cm.
Angles of elevation and depression
Angle of elevation
The angle between the horizontal and the line of sight when looking up at an object.
Angle of depression
The angle between the horizontal and the line of sight when looking down at an object.
Key fact
The angle of depression from A to B equals the angle of elevation from B to A: they are alternate angles between parallel horizontal lines.
Height of a tree
From 20 m away, the angle of elevation of the top of a tree is 35°. The eye is 1.6 m above the ground.
Height above eye level = 20 × tan 35° = 20 × 0.7002 = 14.0 m.
Height of the tree = 14.0 + 1.6 = 15.6 m.
A simple clinometer (a protractor, a straw, a string and a small weight) measures angles of elevation in the field.
Key points to remember
- sin A = opposite ÷ hypotenuse; cos A = adjacent ÷ hypotenuse; tan A = opposite ÷ adjacent.
- Choose the ratio that links the side you know with the side you want.
- sin A = cos (90° - A); tan A = sin A ÷ cos A; sin² A + cos² A = 1.
- Learn the exact values for 30°, 45° and 60°.
- Angles of elevation and depression are measured from the horizontal and are equal for two points looking at each other.
Practice questions
Triangle ABC is right-angled at B, with AB = 8 cm and BC = 15 cm. Find AC, then write sin A, cos A and tan A as fractions. (4 marks)
Show the answer
- AC = sqrt(64 + 225) = sqrt(289) = 17 cm.
- sin A = 15/17, cos A = 8/17, tan A = 15/8.
The hypotenuse of a right-angled triangle is 20 cm and one angle is 32°. Find the side opposite the 32° angle. (2 marks)
Show the answer
20 × sin 32° = 20 × 0.5299 = 10.60 cm.
A ladder 6 m long reaches 5 m up a vertical wall. Find the angle the ladder makes with the ground. (2 marks)
Show the answer
sin A = 5 ÷ 6 = 0.8333, so A = 56.4°.
Given that cos A = 12/13 and A is acute, find sin A and tan A without tables. (3 marks)
Show the answer
- sin² A = 1 - 144/169 = 25/169, so sin A = 5/13.
- tan A = (5/13) ÷ (12/13) = 5/12.
Find x if sin 2x = cos (x + 15°). (2 marks)
Show the answer
2x + x + 15° = 90°, so 3x = 75° and x = 25°.
Without a calculator, evaluate sin 60° × cos 30° + tan 45°. (2 marks)
Show the answer
sqrt(3)/2 × sqrt(3)/2 + 1 = 3/4 + 1 = 1 3/4.
From a point 50 m from the foot of a building, the angle of elevation of the top is 28°. Find the height of the building. (2 marks)
Show the answer
50 × tan 28° = 50 × 0.5317 = 26.59 m.
From the top of a cliff 80 m high, the angles of depression of two boats in line with the cliff are 45° and 30°. Find the distance between the boats. (4 marks)
Show the answer
- Nearer boat: 80 ÷ tan 45° = 80 m from the foot.
- Farther boat: 80 ÷ tan 30° = 80 × 1.7321 = 138.56 m.
- Distance apart = 138.56 - 80 = 58.56 m.
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