This worksheet guides students through finding exact trigonometric values using the properties of special right triangles.

Let’s solve each section step by step.

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Section A: Complete the trig table using special right triangles

We have two special triangles:

1. 30°-60°-90° triangle: sides are 1 (opposite 30°), √3 (opposite 60°), and 2 (hypotenuse)
2. 45°-45°-90° triangle: sides are 1, 1, and √2 (hypotenuse)

Recall:
- sin = opposite / hypotenuse
- cos = adjacent / hypotenuse
- tan = opposite / adjacent

Now fill in the table:

For 30°:
- sin 30° = opposite/hyp = 1/2
- cos 30° = adj/hyp = √3/2
- tan 30° = opp/adj = 1/√3 = √3/3 (rationalized)

For 45°:
- sin 45° = 1/√2 = √2/2
- cos 45° = 1/√2 = √2/2
- tan 45° = 1/1 = 1

For 60°:
- sin 60° = √3/2
- cos 60° = 1/2
- tan 60° = √3/1 = √3

So the completed table is:

| | 0° | 30° | 45° | 60° | 90° |
|-------|------|---------|----------|----------|-----|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | √3/3 | 1 | √3 | — |

(Note: tan 90° is undefined → black box)

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Section B: Give exact values

Use the values from Section A.

1) sin 30 + cos 60 = 1/2 + 1/2 = 1

2) tan 45 + cos 60 = 1 + 1/2 = 3/2

3) 4 sin 30 + cos 0 = 4*(1/2) + 1 = 2 + 1 = 3

4) 2 sin 60 = 2*(√3/2) = √3

5) 8 cos 45 = 8*(√2/2) = 4√2 → 4√2

6) (sin 45)² + 3 cos 60 = (√2/2)² + 3*(1/2) = (2/4) + 3/2 = 1/2 + 3/2 = 2

7) (3 tan 30)² + (tan 60)²
= [3*(√3/3)]² + (√3)²
= (√3)² + 3
= 3 + 3 = 6

8) 5 sin 60 – 3 cos 30
= 5*(√3/2) – 3*(√3/2)
= (5√3 - 3√3)/2 = 2√3/2 = √3

9) 9 sin 30 – 3 sin 90 + 4(cos 45)²
= 9*(1/2) – 3*1 + 4*(√2/2)²
= 9/2 – 3 + 4*(2/4)
= 9/2 – 3 + 4*(1/2)
= 9/2 – 3 + 2
= 9/2 – 1 = 7/2 → 7/2

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Section C: Find missing angles using side ratios

We’ll use inverse trig or recognize special triangle ratios.

1) Triangle with sides: opposite to angle a = ? , adjacent = 3 cm, hypotenuse = 6 cm
Wait — actually, looking at diagram:
Angle “a” is at bottom left. Adjacent side = 3 cm, hypotenuse = 6 cm → so cos(a) = adjacent/hyp = 3/6 = 1/2 → a = 60°

But wait — let’s check: if adjacent = 3, hyp = 6, then it’s a 30-60-90? Actually, in 30-60-90, side opposite 30° is half hypotenuse. Here, side opposite angle a is not given directly.

Actually, better to label:

In triangle 1:
Right angle at bottom right.
Side adjacent to angle a = 3 cm
Hypotenuse = 6 cm
→ So cos(a) = 3/6 = 1/2 → a = 60°

But that would mean angle a is 60°, which makes sense because side opposite would be √(6² - 3²) = √(36-9)=√27=3√3 → which matches 30-60-90 ratio (short leg 3, long leg 3√3, hyp 6). So angle opposite short leg (3) is 30°, so angle a (adjacent to short leg) is 60°. ✔

Answer: a = 60°

2) Triangle: hypotenuse = 10 cm, opposite side to angle b = 5 cm
→ sin(b) = 5/10 = 1/2 → b = 30°

3) Triangle: legs are 2 cm and 2 cm? Wait — diagram shows one leg = 2 cm, hypotenuse = 2√2 cm → this is 45-45-90!
Because in 45-45-90, legs equal, hyp = leg * √2 → here leg = 2, hyp = 2√2 → yes.
Angle c is between leg and hyp → since both legs equal, angles are 45° each → c = 45°

Wait — diagram says: vertical leg = ?, horizontal leg = 2 cm, hyp = 2√2 cm, angle c at top left.
If horizontal leg = 2, hyp = 2√2, then cos(c) = adjacent/hyp = 2/(2√2) = 1/√2 → c = 45° ✔

Answer: c = 45°

4) Triangle: vertical leg = 3√3 cm, horizontal leg = 3 cm, right angle at top left. Angle d at bottom.
So for angle d: opposite = 3√3, adjacent = 3 → tan(d) = (3√3)/3 = √3 → d = 60°

5) Triangle: legs = 4√2 cm and ? , hyp = 8 cm. Right angle at bottom left. Angle e at bottom right.
Adjacent to e = ? , opposite = 4√2, hyp = 8
sin(e) = opposite/hyp = 4√2 / 8 = √2 / 2 → e = 45°

Check: if sin(e) = √2/2, then e=45°, and other leg should also be 4√2 → then hyp = √[(4√2)^2 + (4√2)^2] = √[32+32]=√64=8 → correct.

Answer: e = 45°

6) Triangle: horizontal leg = 6√3 cm, hyp = 12 cm, right angle at top left. Angle f at top right.
For angle f: adjacent = 6√3, hyp = 12 → cos(f) = 6√3 / 12 = √3 / 2 → f = 30°

Check: opposite side = √(12² - (6√3)²) = √(144 - 108) = √36 = 6 → so sides 6, 6√3, 12 → 30-60-90. Angle f adjacent to long leg → so f = 30° ✔

Answer: f = 30°

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Final Answer:

Section A Table:

| | 0° | 30° | 45° | 60° | 90° |
|-------|------|---------|----------|----------|-----|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | √3/3 | 1 | √3 | — |

Section B Answers:

1) 1
2) 3/2
3) 3
4) √3
5) 4√2
6) 2
7) 6
8) √3
9) 7/2

Section C Answers:

1) a = 60°
2) b = 30°
3) c = 45°
4) d = 60°
5) e = 45°
6) f = 30°