Unit 6: Introduction to Trigonometry
Exercise 6.4 Solutions
Trigonometric values of special angles (30°, 45°, 60°) and evaluating numeric expressions using the special angle table.
Special Angles Reference Table
Key Values
These are the fundamental exact values derived from the 30-60-90 and 45-45-90 triangles:
| Ratio | 0° | 30° = π/6 | 45° = π/4 | 60° = π/3 | 90° |
|---|---|---|---|---|---|
| sin | 0 | 12 | 1√2 | √32 | 1 |
| cos | 1 | √32 | 1√2 | 12 | 0 |
| tan | 0 | 1√3 | 1 | √3 | Undefined |
| cosec | Undefined | 2 | √2 | 2√3 | 1 |
| sec | 1 | 2√3 | √2 | 2 | Undefined |
| cot | Undefined | √3 | 1 | 1√3 | 0 |
Question 1
Special Angle Values
Find the value of the following trigonometric ratios without using a calculator:
Solution
(i) sin 30°
sin 30° = 12
sin 30° = 12
(ii) cos 30°
cos 30° = √32
cos 30° = √32
(iii) tan π/6
tan π/6 = tan 30° = 1√3
tan π/6 = tan 30° = 1√3
(iv) tan 60°
tan 60° = √3
tan 60° = √3
(v) sec 60°
sec 60° = 1cos 60° = 112 = 2
sec 60° = 1cos 60° = 112 = 2
(vi) cos π/3
cos π/3 = cos 60° = 12
cos π/3 = cos 60° = 12
(vii) cot 60°
cot 60° = 1tan 60° = 1√3
cot 60° = 1tan 60° = 1√3
(viii) sin 60°
sin 60° = √32
sin 60° = √32
(ix) sec 30°
sec 30° = 1cos 30° = 1√32 = 2√3
sec 30° = 1cos 30° = 1√32 = 2√3
(x) cosec 30°
cosec 30° = 1sin 30° = 112 = 2
cosec 30° = 1sin 30° = 112 = 2
(xi) sin 45°
sin 45° = 1√2
sin 45° = 1√2
(xii) cos π/4
cos π/4 = cos 45° = 1√2
cos π/4 = cos 45° = 1√2
Question 2
Evaluate Expressions
Evaluate the following expressions using special angle values:
Solution
(i) 2 sin 60° cos 60°
= 2 × √32 × 12
= 2√34 = √32
= 2√34 = √32
(ii) 2 cos (π/6) sin (π/6)
= 2 cos 30° sin 30°
= 2 × √32 × 12
= 2√34 = √32
= 2 × √32 × 12
= 2√34 = √32
(iii) 2 sin 45° + 2 cos 45°
= 2 × 1√2 + 2 × 1√2
= 2√2 + 2√2 = 4√2
Rationalising: = 4√2 × √2√2 = 4√22 = 2√2
= 2√2 + 2√2 = 4√2
Rationalising: = 4√2 × √2√2 = 4√22 = 2√2
(iv) sin 60° cos 30° + cos 60° sin 30°
= √32 × √32 + 12 × 12
= 34 + 14 = 44 = 1
= 34 + 14 = 44 = 1
Note
This result confirms the Addition Formula: sin(A + B) = sin A cos B + cos A sin B. Here A = 60°, B = 30°, so sin(90°) = 1. ✓
(v) cos 60° cos 30° − sin 60° sin 30°
= 12 × √32 − √32 × 12
= √34 − √34 = 0
= √34 − √34 = 0
Note
This confirms: cos(A + B) = cos A cos B − sin A sin B. Here A + B = 60° + 30° = 90°, and cos 90° = 0. ✓
(vi) sin 60° cos 30° − cos 60° sin 30°
= √32 × √32 − 12 × 12
= 34 − 14 = 24 = 12
= 34 − 14 = 24 = 12
Note
This confirms: sin(A − B) = sin A cos B − cos A sin B. Here A − B = 60° − 30° = 30°, and sin 30° = 1/2. ✓
(vii) cos 60° cos 30° + sin 60° sin 30°
= 12 × √32 + √32 × 12
= √34 + √34 = 2√34 = √32
= √34 + √34 = 2√34 = √32
Note
This confirms: cos(A − B) = cos A cos B + sin A sin B. Here A − B = 60° − 30° = 30°, and cos 30° = √3/2. ✓
(viii) tan (π/6) + cot (π/6)
= tan 30° + cot 30°
= 1√3 + √3
= 1√3 + √3 × √3√3 = 1 + 3√3 = 4√3 = 4√33
= 1√3 + √3
= 1√3 + √3 × √3√3 = 1 + 3√3 = 4√3 = 4√33
Mathematical Note – Textbook Typo
The textbook question (viii) prints "tan(π/6) cot(π/6)" which could imply multiplication (= 1), but the textbook solved key uses addition and obtains 4/√3. The correct interpretation is tan(π/6) + cot(π/6) = 4√3/3, which is what we have solved here.
Question 3
Mixed Expressions
If sin 45° and cos 45° are each equal to 1√2, find the value of:
Solution
(i) 2 sin 45° − 2 cos 45°
= 2 × 1√2 − 2 × 1√2
= 2√2 − 2√2 = 0
= 2√2 − 2√2 = 0
(ii) 3 cos 45° + 4 sin 30°
= 3 × 1√2 + 4 × 12
= 3√2 + 2
Rationalising: = 3√22 + 2 = 3√2 + 42 ≈ 4.121
= 3√2 + 2
Rationalising: = 3√22 + 2 = 3√2 + 42 ≈ 4.121
Mathematical Note – Textbook Typo (Q.3 ii)
The textbook key's solution step evaluates "3 cos 45° + 4 sin 45°" (using sin 45° = 1/√2), which gives 7/√2 = 7√2/2. However, the question text clearly states "sin 30°". Both versions are solved here for completeness. Using sin 45° gives: 3/√2 + 4/√2 = 7/√2 = 7√2/2 ≈ 4.95.
(iii) 5 cos 45° − 3 sin 45°
= 5 × 1√2 − 3 × 1√2
= 5 − 3√2 = 2√2
Rationalising: = 2√2 × √2√2 = 2√22 = √2
= 5 − 3√2 = 2√2
Rationalising: = 2√2 × √2√2 = 2√22 = √2
🎲 Special Angles Expression Evaluator
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