Trigonometri AS/A level puremath 1
Comprehensive Guide to Trigonometry: A/AS Level
Trigonometry is a fundamental branch of mathematics that deals with the relationships between the sides and angles of triangles. This comprehensive guide covers key concepts, formulas, and applications of trigonometry at the A/AS level.
1. Basic Trigonometric Ratios
The three primary trigonometric ratios are sine, cosine, and tangent. These are defined in terms of the sides of a right-angled triangle:
For an angle θ in a right-angled triangle:
- $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$$
- $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$$
- $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}$$
2. Important Trigonometric Identities
Several key identities are essential in trigonometry:
- $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ (when cos θ ≠ 0)
- $$\sin^2 \theta + \cos^2 \theta = 1$$ (Pythagorean identity)
3. Exact Values of Trigonometric Functions
Memorizing these exact values is crucial for solving trigonometric problems efficiently:
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° = π/6 | 1/2 | √3/2 | 1/√3 |
| 45° = π/4 | 1/√2 | 1/√2 | 1 |
| 60° = π/3 | √3/2 | 1/2 | √3 |
4. Positive and Negative Angles
Understanding the signs of trigonometric functions in different quadrants is essential:
- Positive angles are measured counterclockwise from the positive x-axis
- Negative angles are measured clockwise from the positive x-axis
Remember the mnemonic "All Students Take Calculus" to recall where each function is positive.
5. Graphs of Trigonometric Functions
Understanding the graphs of sine, cosine, and tangent is crucial for solving equations and inequalities:
6. Transformations of Trigonometric Functions
Understanding how parameters affect trigonometric graphs is essential:
- $$y = a \sin x$$ : Vertical stretch by factor a
- $$y = \sin (ax)$$ : Horizontal compression by factor 1/a
- $$y = a + \sin x$$ : Vertical translation by a units
- $$y = \sin (x + a)$$ : Horizontal translation by -a units
7. Inverse Trigonometric Functions
Inverse functions are crucial for solving equations:
- $$y = \sin^{-1} x$$ : Domain: [-1, 1], Range: [-π/2, π/2]
- $$y = \cos^{-1} x$$ : Domain: [-1, 1], Range: [0, π]
- $$y = \tan^{-1} x$$ : Domain: All real numbers, Range: (-π/2, π/2)
8. Example Problems
Problem 1: Solve the equation $$\sin x = \frac{1}{2}$$ for 0 ≤ x < 2π.
Solution:
- We know that $$\sin 30° = \frac{1}{2}$$
- Therefore, x = 30° = π/6 is one solution
- The other solution is 180° - 30° = 150° = 5π/6
- Thus, the solutions are x = π/6 and x = 5π/6
Problem 2: Find the exact value of $$\tan (60° + 45°)$$.
Solution:
- We can use the tangent addition formula: $$\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$
- $$\tan 60° = \sqrt{3}$$ and $$\tan 45° = 1$$
- Substituting: $$\tan (60° + 45°) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}$$
- Rationalizing the denominator: $$\frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}$$
Conclusion
Mastering these trigonometric concepts and techniques is essential for success in A/AS level mathematics. Regular practice with a variety of problems will help solidify your understanding and improve your problem-solving skills.
Remember to explore additional resources and practice problems to further enhance your trigonometry knowledge. Good luck with your studies!
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