Trigonometry Special Angles
Learn the values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°
Special angles (0°, 30°, 45°, 60°, 90°) are the building blocks of trigonometry. Memorizing these values will help you solve problems quickly without a calculator.
What You'll Learn
Quiz 1: Basic Ratios
What is sin 30°?
1. Trigonometric Ratios
Sine (sin θ)
- Formula: Opposite / Hypotenuse
- sin θ = Opposite ÷ Hypotenuse
Cosine (cos θ)
- Formula: Adjacent / Hypotenuse
- cos θ = Adjacent ÷ Hypotenuse
Tangent (tan θ)
- Formula: Opposite / Adjacent
- tan θ = Opposite ÷ Adjacent
- tan θ = sin θ / cos θ
Opposite = side opposite the angle θ
Adjacent = side next to angle θ (not the hypotenuse)
Hypotenuse = longest side, opposite the right angle
Quiz 2: 45° Values
What is tan 45°?
2. How We Get Special Angles
From Equilateral Triangle
- Start with equilateral triangle (all sides = 2)
- Bisect it → two 30-60-90 triangles
- Sides: 1 (short), √3 (medium), 2 (hypotenuse)
From Isosceles Right Triangle
- Two equal sides of length 1
- Hypotenuse = √2 (by Pythagoras)
- Angles: 45°, 45°, 90°
From Unit Circle
- Circle with radius = 1
- Point on circle: (cos θ, sin θ)
- At 0°: (1, 0); at 90°: (0, 1)
3. Complete Values Table
| Angle (θ) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
sin values: √0/2, √1/2, √2/2, √3/2, √4/2 → 0, 1/2, √2/2, √3/2, 1
cos values are the reverse order of sin
Quiz 3: 60° Values
What is cos 60°?
4. Example Problems
Example 1: Basic Evaluation
Find the value of: sin 30° + cos 60°
- sin 30° = 1/2
- cos 60° = 1/2
- Add: 1/2 + 1/2 = 1
Example 2: Simplify Expression
Simplify: (tan 45° × sin 60°) ÷ cos 30°
- tan 45° = 1, sin 60° = √3/2, cos 30° = √3/2
- Expression: (1 × √3/2) ÷ (√3/2)
- = (√3/2) × (2/√3) = 1
Example 3: Solve Equation
Solve for x: 2 × sin x = 1, where 0° ≤ x ≤ 90°
- Divide both sides by 2: sin x = 1/2
- From the table, sin 30° = 1/2
- Check domain: 30° is between 0° and 90°
Quiz 4: Solve Equation
Solve: 2 × tan x = 2, where 0° ≤ x ≤ 90°
5. Key Relationships
Complementary Angles
- sin 30° = cos 60° = 1/2
- sin 60° = cos 30° = √3/2
- sin 45° = cos 45° = √2/2
- sin θ = cos(90° - θ)
Pythagorean Identity
- sin²θ + cos²θ = 1
- Check: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1
- Check: (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1
tan θ Relationship
- tan θ = sin θ / cos θ
- tan 30° = (1/2) ÷ (√3/2) = 1/√3 = √3/3
- tan 45° = (√2/2) ÷ (√2/2) = 1
- tan 60° = (√3/2) ÷ (1/2) = √3
Quiz 5: Complementary Angles
sin 30° is equal to:
6. Common Mistakes
Confusing sin 30° and sin 60° - remember sin 30° = 1/2, sin 60° = √3/2
tan 90° is undefined, not 0 or infinity - it's not a real number
Forgetting to rationalize denominators - write 1/√2 as √2/2
Wrong triangle labeling - opposite is across from angle, adjacent is next to angle
Quiz 6: 30° and 60°
Which statement is TRUE?
7. Practice Questions
Find: sin 45° + cos 45°
Simplify: tan 60° × sin 30°
Solve: cos x = √3/2, 0° ≤ x ≤ 90°
8. Summary
Values to Memorize
- sin 0° = 0, cos 0° = 1, tan 0° = 0
- sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
- sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
- sin 90° = 1, cos 90° = 0, tan 90° = undefined
Quick Reference
- sin 30° = cos 60° = 1/2
- sin 60° = cos 30° = √3/2
- sin 45° = cos 45° = √2/2
- tan 45° = 1
- sin²θ + cos²θ = 1
