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📏 Trigonometry

Grade 9 · Cambridge IGCSE · Right-Angled Triangles

Labelling Sides

Opposite, Adjacent, Hypotenuse relative to angle θ

SOH CAH TOA

The three trig ratios for right-angled triangles

Finding Sides

Rearrange the trig ratio to find a missing side

Finding Angles

Use inverse trig sin⁻¹, cos⁻¹, tan⁻¹

Exact Values

sin 30°, cos 60°, tan 45°, tan 60°

Elevation & Bearings

Angles of elevation, depression, bearings

1. Labelling Sides

For a given angle θ in a right-angled triangle, the three sides are always:

Hypotenuse (H): The longest side — opposite the right angle. Always the same regardless of which angle you choose.
Opposite (O): The side directly opposite angle θ.
Adjacent (A): The side next to angle θ (but not the hypotenuse).
The labels O, A, H change depending on which angle you are working with. H is always fixed, but O and A switch.

2. SOH CAH TOA

The three trigonometric ratios for right-angled triangles:

SOH
Sine
sin θ = O/H
CAH
Cosine
cos θ = A/H
TOA
Tangent
tan θ = O/A
Memory aid: Some Old Hens Can Always Hide Their Old Age — or simply SOH CAH TOA.

3. Finding a Missing Side

Choose the ratio that connects the angle, the known side and the unknown side. Then rearrange:

To find Opp: O = H × sin θ  |  O = A × tan θ
To find Adj: A = H × cos θ  |  A = O ÷ tan θ
To find Hyp: H = O ÷ sin θ  |  H = A ÷ cos θ
Example: θ = 30°, H = 10. Find O.
sin 30° = O/10 → O = 10 × sin 30° = 10 × 0.5 = 5

4. Finding a Missing Angle

When two sides are known, use the inverse trig function to find the angle:

θ = sin⁻¹(O/H)  |  θ = cos⁻¹(A/H)  |  θ = tan⁻¹(O/A)
Example: O = 5, H = 10. Find θ.
θ = sin⁻¹(5/10) = sin⁻¹(0.5) = 30°
Make sure your calculator is in degrees mode (DEG or D) not radians.

5. Exact Values

These values must be memorised for IGCSE exams:

sin 30° = 0.5    cos 30° = √3/2 ≈ 0.866
sin 45° = cos 45° = √2/2 ≈ 0.707
sin 60° = √3/2 ≈ 0.866    cos 60° = 0.5
tan 30° = 1/√3 ≈ 0.577    tan 45° = 1    tan 60° = √3 ≈ 1.732

6. Angles of Elevation and Depression

Angle of elevation: The angle measured upward from the horizontal to the line of sight.
Angle of depression: The angle measured downward from the horizontal to the line of sight.
Example: From a point 50 m from the base of a tower, the angle of elevation to the top is 40°. Find the height.
tan 40° = height / 50 → height = 50 × tan 40° ≈ 50 × 0.839 = 42.0 m

7. Bearings

Bearings are measured clockwise from North, always given as 3 digits (e.g., 045°, 270°).
Trig can be used to find the horizontal/vertical components of a journey on a bearing.
Example: A plane flies 200 km on bearing 060°. How far east and north has it travelled?
East = 200 × sin 60° ≈ 173 km    North = 200 × cos 60° = 100 km

Example 1 — Find a Side Using Sin

θ = 30°, hypotenuse = 10 cm. Find the opposite side.

Choose ratio: sin θ = opp/hyp → opp = hyp × sin θ
Substitute: opp = 10 × sin 30° = 10 × 0.5 = 5 cm

Example 2 — Find a Side Using Cos

θ = 30°, hypotenuse = 10 cm. Find the adjacent side.

Choose ratio: cos θ = adj/hyp → adj = hyp × cos θ
Substitute: adj = 10 × cos 30° = 10 × (√3/2) ≈ 8.66 cm

Example 3 — Find a Side Using Tan

θ = 30°, adjacent = 10 cm. Find the opposite side.

Choose ratio: tan θ = opp/adj → opp = adj × tan θ
Substitute: opp = 10 × tan 30° = 10 × (1/√3) ≈ 5.77 cm

Example 4 — Find an Angle

Opposite = 5, hypotenuse = 10. Find θ.

Choose ratio: sin θ = opp/hyp = 5/10 = 0.5
Inverse: θ = sin⁻¹(0.5) = 30°

Example 5 — Angle of Elevation

From a point 40 m from a building, the angle of elevation to the roof is 35°. Find the building height.

Setup: tan 35° = height / 40
Rearrange: height = 40 × tan 35° ≈ 40 × 0.700 ≈ 28.0 m

Example 6 — Find Hypotenuse

θ = 45°, opposite = 7 cm. Find the hypotenuse.

Choose ratio: sin 45° = opp/hyp → hyp = opp/sin 45°
Substitute: hyp = 7 / (√2/2) = 7√2 ≈ 9.90 cm

📐 Trig Calculator

Choose what you know and what to find, enter values, and see the triangle drawn with working.

Select mode and enter values.

Exercise 1 — Find a Side Using Sin (2 d.p.)

1. θ=30°, hyp=10. Find opp.

2. θ=45°, hyp=10. Find opp.

3. θ=60°, hyp=10. Find opp.

4. θ=35°, hyp=10. Find opp.

5. θ=30°, hyp=20. Find opp.

6. θ=60°, hyp=5. Find opp.

7. θ=45°, hyp=5. Find opp.

8. θ=67°, hyp=10. Find opp.

9. θ=40°, hyp=10. Find opp.

10. θ=50°, hyp=16.8. Find opp.

Exercise 2 — Find a Side Using Cos (2 d.p.)

1. θ=30°, hyp=10. Find adj.

2. θ=45°, hyp=10. Find adj.

3. θ=60°, hyp=10. Find adj.

4. θ=0°, hyp=10. Find adj.

5. θ=40°, hyp=10. Find adj.

6. θ=20°, hyp=10. Find adj.

7. θ=10°, hyp=10. Find adj.

8. θ=50°, hyp=10. Find adj.

9. θ=35°, hyp=10. Find adj.

10. θ=65°, hyp=10. Find adj.

Exercise 3 — Find a Side Using Tan (2 d.p.)

1. θ=30°, adj=10. Find opp.

2. θ=45°, adj=10. Find opp.

3. θ=30°, adj=8.66. Find opp. [≈5]

4. θ=60°, adj=10. Find opp.

5. θ=40°, adj=10. Find opp.

6. θ=20°, adj=10. Find opp.

7. θ=35°, adj=10. Find opp.

8. θ=50°, adj=10. Find opp.

9. θ=22°, adj=9.9. Find opp.

10. θ=63°, adj=10. Find opp.

Exercise 4 — Find the Angle in Degrees (1 d.p.)

1. sin θ = 0.5. Find θ.

2. cos θ = 0.707 (=1/√2). Find θ.

3. tan θ = √3. Find θ.

4. opp=3, hyp=5. Find θ (sin⁻¹).

5. adj=3, hyp=5. Find θ (cos⁻¹).

6. opp=4, adj=4.5. Find θ (tan⁻¹).

7. opp=7, adj=6.3. Find θ (tan⁻¹).

8. opp=3, adj=6. Find θ (tan⁻¹).

9. opp=9, adj=4.5. Find θ (tan⁻¹).

10. opp=5, adj=7.5. Find θ (tan⁻¹).

Exercise 5 — Elevation & Bearing Problems (1 d.p.)

1. From 100 m away, angle of elevation to top = 30°. Height of building?

2. From 50 m away, angle of elevation = 45°. Height?

3. Ladder 10 m at angle 60° to ground. How high up the wall?

4. Cliff is 120 m high. Angle of depression to boat = 45°. Distance from cliff base to boat?

5. Tower 80 m. Angle of depression from top to car = 63.4°. Distance from base to car? [tan⁻¹(4)]

6. Plane flies 500 km on bearing 038°. How far north has it travelled?

7. Ship sails 200 km on bearing 051°. How far east?

8. Height of tree = 15 m. Angle of elevation from observer = 36.9°. How far is observer from tree base?

9. Angle of elevation of sun = 53.1°. Pole 12 m tall. Shadow length?

10. From top of 50 m cliff, angle of depression to ship = 22.6°. Distance of ship from base?

🏋️ Practice — 20 Questions

1. sin θ=opp/hyp: θ=30°, hyp=10. Find opp.

2. θ=45°, hyp=10. Find opp (sin).

3. θ=60°, hyp=10. Find opp (sin).

4. θ=35°, hyp=10. Find opp (sin).

5. θ=30°, hyp=20. Find opp (sin).

6. θ=60°, hyp=5. Find opp (sin).

7. θ=45°, hyp=5. Find opp (sin).

8. θ=67°, hyp=10. Find opp (sin).

9. θ=40°, hyp=10. Find opp (sin).

10. θ=50°, hyp=16.8. Find opp (sin).

11. θ=30°, hyp=10. Find adj (cos).

12. θ=45°, hyp=10. Find adj (cos).

13. θ=60°, hyp=10. Find adj (cos).

14. θ=0°, hyp=10. Find adj (cos).

15. θ=40°, hyp=10. Find adj (cos).

16. θ=20°, hyp=10. Find adj (cos).

17. θ=10°, hyp=10. Find adj (cos).

18. θ=50°, hyp=10. Find adj (cos).

19. θ=35°, hyp=10. Find adj (cos).

20. θ=65°, hyp=10. Find adj (cos).

🏆 Challenge — 8 Questions

1. sin θ = 0.5. Find θ (degrees).

2. cos θ ≈ 0.707. Find θ (degrees).

3. tan θ = √3 ≈ 1.732. Find θ (degrees).

4. opp=3, hyp=5. Find θ using sin⁻¹ (1 d.p.).

5. adj=3, hyp=5. Find θ using cos⁻¹ (1 d.p.).

6. opp=4, adj=4.5. Find θ using tan⁻¹ (1 d.p.).

7. opp=7, adj=6.3. Find θ using tan⁻¹ (1 d.p.).

8. opp=3, adj=6. Find θ using tan⁻¹ (1 d.p.).