Trigonometry
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Convert Between Radians and Degrees
Conversion Formulas:
Degrees to Radians:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
Radians to Degrees:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]
Key Equivalences:
| Degrees | Radians |
|---|---|
| 360° | \( 2\pi \) |
| 180° | \( \pi \) |
| 90° | \( \frac{\pi}{2} \) |
| 60° | \( \frac{\pi}{3} \) |
| 45° | \( \frac{\pi}{4} \) |
| 30° | \( \frac{\pi}{6} \) |
2. Radians and Arc Length
Arc Length Formula:
\[ s = r\theta \]
• \( s \) = arc length
• \( r \) = radius of circle
• \( \theta \) = central angle in RADIANS
⚠️ Angle must be in radians, not degrees!
3. Quadrants
Four Quadrants:
| Quadrant | Angle Range | Signs |
|---|---|---|
| I | 0° to 90° or \( 0 \) to \( \frac{\pi}{2} \) | All positive |
| II | 90° to 180° or \( \frac{\pi}{2} \) to \( \pi \) | sin (+), cos (−), tan (−) |
| III | 180° to 270° or \( \pi \) to \( \frac{3\pi}{2} \) | tan (+), sin (−), cos (−) |
| IV | 270° to 360° or \( \frac{3\pi}{2} \) to \( 2\pi \) | cos (+), sin (−), tan (−) |
4. Coterminal and Reference Angles
Coterminal Angles:
Angles that share the same terminal side
\[ \theta + 360°n \text{ or } \theta + 2\pi n \quad (n \in \mathbb{Z}) \]
Reference Angle:
The acute angle formed between the terminal side and the x-axis
• Quadrant I: \( \theta' = \theta \)
• Quadrant II: \( \theta' = 180° - \theta \) or \( \pi - \theta \)
• Quadrant III: \( \theta' = \theta - 180° \) or \( \theta - \pi \)
• Quadrant IV: \( \theta' = 360° - \theta \) or \( 2\pi - \theta \)
5. Trigonometric Ratios (Right Triangles)
Six Trigonometric Functions:
\[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \csc\theta = \frac{\text{hypotenuse}}{\text{opposite}} \quad \sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}} \quad \cot\theta = \frac{\text{adjacent}}{\text{opposite}} \]
Reciprocal Identities:
\[ \csc\theta = \frac{1}{\sin\theta} \quad \sec\theta = \frac{1}{\cos\theta} \quad \cot\theta = \frac{1}{\tan\theta} \]
6. Unit Circle and Special Angles
Unit Circle Definition:
Circle with radius 1 centered at origin. Point \( (x, y) \) on the circle corresponds to:
\[ \cos\theta = x \quad \sin\theta = y \quad \tan\theta = \frac{y}{x} \]
Special Angles (30°, 45°, 60°):
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° or 0 | 0 | 1 | 0 |
| 30° or \( \frac{\pi}{6} \) | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{3}}{3} \) |
| 45° or \( \frac{\pi}{4} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{2}}{2} \) | 1 |
| 60° or \( \frac{\pi}{3} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) |
| 90° or \( \frac{\pi}{2} \) | 1 | 0 | undefined |
7. Inverse Trigonometric Functions
Inverse Functions:
• \( \sin^{-1}(x) \) or \( \arcsin(x) \): Returns angle whose sine is x
• \( \cos^{-1}(x) \) or \( \arccos(x) \): Returns angle whose cosine is x
• \( \tan^{-1}(x) \) or \( \arctan(x) \): Returns angle whose tangent is x
Domain and Range:
| Function | Domain | Range |
|---|---|---|
| \( \sin^{-1}(x) \) | [-1, 1] | \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) |
| \( \cos^{-1}(x) \) | [-1, 1] | \( [0, \pi] \) |
| \( \tan^{-1}(x) \) | All real numbers | \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) |
8. Solve Trigonometric Equations
Steps:
1. Isolate the trigonometric function
2. Use inverse functions or special angles
3. Find all solutions in the given interval
4. Consider all quadrants where the function has that value
9. Law of Sines
Formula:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
where a, b, c are sides and A, B, C are opposite angles
When to Use:
• AAS (Angle-Angle-Side)
• ASA (Angle-Side-Angle)
• SSA (Side-Side-Angle) - ambiguous case
10. Law of Cosines
Formulas:
\[ a^2 = b^2 + c^2 - 2bc\cos A \] \[ b^2 = a^2 + c^2 - 2ac\cos B \] \[ c^2 = a^2 + b^2 - 2ab\cos C \]
When to Use:
• SAS (Side-Angle-Side)
• SSS (Side-Side-Side)
11. Area of a Triangle
Sine Formula:
\[ \text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B \]
Use when you know two sides and the included angle
Heron's Formula:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
\[ \text{where } s = \frac{a+b+c}{2} \text{ (semi-perimeter)} \]
Use when you know all three sides
12. Quick Reference - Key Identities
Pythagorean Identities:
\( \sin^2\theta + \cos^2\theta = 1 \)
\( 1 + \tan^2\theta = \sec^2\theta \)
\( 1 + \cot^2\theta = \csc^2\theta \)
Quotient Identities:
\( \tan\theta = \frac{\sin\theta}{\cos\theta} \)
\( \cot\theta = \frac{\cos\theta}{\sin\theta} \)
📚 Study Tips
✓ Memorize special angle values (30°, 45°, 60°)
✓ For arc length formula, angle MUST be in radians
✓ Use Law of Sines for AAS/ASA; Law of Cosines for SAS/SSS
✓ Remember CAST rule for signs in quadrants (All-Sin-Tan-Cos)
✓ Reference angles are always between 0° and 90°