Trigonometry formulas for Mathematics

 Here are the trigonometry formulas commonly encountered in mathematics:

1. Basic Trigonometric Ratios:

- Sine: `\( \sin(\theta) = \frac{{\text{opposite}}}{{\text{hypotenuse}}} \)`

- Cosine: `\( \cos(\theta) = \frac{{\text{adjacent}}}{{\text{hypotenuse}}} \)`

- Tangent: `\( \tan(\theta) = \frac{{\text{opposite}}}{{\text{adjacent}}} \)`

2. Reciprocal Trigonometric Ratios:

- Cosecant: `\( \csc(\theta) = \frac{1}{{\sin(\theta)}} = \frac{{\text{hypotenuse}}}{{\text{opposite}}} \)`

- Secant: `\( \sec(\theta) = \frac{1}{{\cos(\theta)}} = \frac{{\text{hypotenuse}}}{{\text{adjacent}}} \)`

- Cotangent: `\( \cot(\theta) = \frac{1}{{\tan(\theta)}} = \frac{{\text{adjacent}}}{{\text{opposite}}} \)`

3. Pythagorean Identities:

- `\( \sin^2(\theta) + \cos^2(\theta) = 1 \)`

- `\( 1 + \tan^2(\theta) = \sec^2(\theta) \)`

- `\( 1 + \cot^2(\theta) = \csc^2(\theta) \)`

4. Angle Sum and Difference Identities:

- `\( \sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) \)`

- `\( \cos(A \pm B) = \cos(A)\cos(B) \pm \sin(A)\sin(B) \)`

- `\( \tan(A \pm B) = \frac{{\tan(A) \pm \tan(B)}}{{1 \mp \tan(A)\tan(B)}} \)`

5. Double Angle Identities:

- `\( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)`

- `\( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta) \)`

- `\( \tan(2\theta) = \frac{{2\tan(\theta)}}{{1 - \tan^2(\theta)}} \)`

6. Half Angle Identities:

- `\( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{{1 - \cos(\theta)}}{2}} \)`

- `\( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{{1 + \cos(\theta)}}{2}} \)`

- `\( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{{1 - \cos(\theta)}}{{1 + \cos(\theta)}}} \)`

7. Law of Sines:

-` \( \frac{{a}}{{\sin(A)}} = \frac{{b}}{{\sin(B)}} = \frac{{c}}{{\sin(C)}} \)`

8. Law of Cosines:

- `\( a^2 = b^2 + c^2 - 2bc\cos(A) \)`

- `\( b^2 = a^2 + c^2 - 2ac\cos(B) \)`

- `\( c^2 = a^2 + b^2 - 2ab\cos(C) \)`

9. Product-to-Sum and Sum-to-Product Identities:

- Product-to-Sum:

  - `\( \sin(A)\sin(B) = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \)`

  - `\( \cos(A)\cos(B) = \frac{1}{2}[\cos(A-B) + \cos(A+B)] \)`

  - `\( \sin(A)\cos(B) = \frac{1}{2}[\sin(A-B) + \sin(A+B)] \)`

- Sum-to-Product:

  - `\( \sin(A) + \sin(B) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \)`

  - `\( \sin(A) - \sin(B) = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \)`

  - `\( \cos(A) + \cos(B) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \)`

  - `\( \cos(A) - \cos(B) = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \)`

10. Area of a Triangle Using Trigonometry:

- If `\( a \), \( b \), and \( c \)` are the lengths of the sides of a triangle, and `\( A \)` is the angle opposite side `\( a \),` then the area of the triangle is given by:

  - `\( \text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2}bc\sin(A) = \frac{1}{2}ca\sin(B) \)`

11. Sine and Cosine Addition Formulas:

- `\( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)`

- `\( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)`

12. Sine and Cosine Half-Angle Formulas:

- `\( \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2} \)`

- `\( \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2} \)`

13. Cofunction Identities:

- `\( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \)`

- `\( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \)`

- `\( \tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right) \)`

14. Law of Tangents:

- `\( \frac{{a - b}}{{a + b}} = \frac{{\tan\left(\frac{{A - B}}{2}\right)}}{{\tan\left(\frac{{A + B}}{2}\right)}} \)`

15. Half-Angle Tangent Formula:

- `\( \tan\left(\frac{\theta}{2}\right) = \frac{{\sin(\theta)}}{{1 + \cos(\theta)}} \)` or `\( \tan\left(\frac{\theta}{2}\right) = \frac{{1 - \cos(\theta)}}{{\sin(\theta)}} \)`

16. Angle Conversion Formulas:

- Degrees to Radians: `\( \text{Radians} = \frac{{\text{Degrees} \times \pi}}{{180}} \)`

- Radians to Degrees: `\( \text{Degrees} = \frac{{\text{Radians} \times 180}}{{\pi}} \)`

17. Periodicity Formulas:

- `\( \sin(\theta + 2\pi) = \sin(\theta) \)`

- `\( \cos(\theta + 2\pi) = \cos(\theta) \)`

- `\( \tan(\theta + \pi) = \tan(\theta) \)`

18. Inverse Trigonometric Functions:

- `\( \sin^{-1}(x) = \arcsin(x) \), where \( -\frac{\pi}{2} \leq \arcsin(x) \leq \frac{\pi}{2} \)`

- `\( \cos^{-1}(x) = \arccos(x) \), where \( 0 \leq \arccos(x) \leq \pi \)`

- `\( \tan^{-1}(x) = \arctan(x) \),` 

`where \( -\frac{\pi}{2} < \arctan(x) < \frac{\pi}{2} \)`

19. Trigonometric Identities Involving Powers:

- `\( \sin^3(\theta) = \frac{3}{4}\sin(\theta) - \frac{1}{4}\sin(3\theta) \)`

- `\( \cos^3(\theta) = \frac{3}{4}\cos(\theta) + \frac{1}{4}\cos(3\theta) \)`

20. Trigonometric Functions of Negative Angles:

- `\( \sin(-\theta) = -\sin(\theta) \)`

- `\( \cos(-\theta) = \cos(\theta) \)`

- `\( \tan(-\theta) = -\tan(\theta) \)`

21. Trigonometric Addition and Subtraction Formulas:

- `\( \sin(A + B) = \sin A \cos B + \cos A \sin B \)`

- `\( \sin(A - B) = \sin A \cos B - \cos A \sin B \)`

- `\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)`

- `\( \cos(A - B) = \cos A \cos B + \sin A \sin B \)`

22. Trigonometric Triple Angle Formulas:

- `\( \sin(3\theta) = 3\sin\theta - 4\sin^3\theta \)`

- `\( \cos(3\theta) = 4\cos^3\theta - 3\cos\theta \)`

- `\( \tan(3\theta) = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} \)`

23. Trigonometric Half-Angle Formulas:

- `\( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}} \)`

- `\( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}} \)`

- `\( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} \)`

24. Trigonometric Sum and Difference of Cubes Formulas:

- `\( \sin(A + B)\sin(A - B) = \sin^2A - \sin^2B \)`

- `\( \cos(A + B)\cos(A - B) = \cos^2A - \cos^2B \)`

25. Euler's Formula:

- `\( e^{i\theta} = \cos\theta + i\sin\theta \)`

26. Hyperbolic Trigonometric Formulas:

- `\( \sinh(x) = \frac{e^x - e^{-x}}{2} \)`

- `\( \cosh(x) = \frac{e^x + e^{-x}}{2} \)`

- `\( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)`

27. Inverse Hyperbolic Trigonometric Functions:

- `\( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \)`

- `\( \cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \)`

- `\( \tanh^{-1}(x) = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right) \)`

28. Power-Reducing Formulas:

- `\( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \)`

- `\( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \)`

- `\( \tan^2(\theta) = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)} \)`

29. Law of Sines for Angles:

- `\( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \)`

30. Law of Cosines for Angles:

- `\( \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \)`

- `\( \cos(B) = \frac{c^2 + a^2 - b^2}{2ca} \)`

- `\( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \)`

31. Polar Coordinates Conversion:

- `\( x = r \cos(\theta) \)`

- `\( y = r \sin(\theta) \)`

32. Trigonometric Integration Formulas:

- `\( \int \sin(x) \, dx = -\cos(x) + C \)`

- `\( \int \cos(x) \, dx = \sin(x) + C \)`

- `\( \int \tan(x) \, dx = -\ln|\cos(x)| + C \)`

33. Trigonometric Differentiation Formulas:

- `\( \frac{d}{dx}[\sin(x)] = \cos(x) \)`

- `\( \frac{d}{dx}[\cos(x)] = -\sin(x) \)`

- `\( \frac{d}{dx}[\tan(x)] = \sec^2(x) \)`

34. Trigonometric Identities for Even and Odd Functions:

- `\( \sin(-x) = -\sin(x) \) (Odd Function)`

- `\( \cos(-x) = \cos(x) \) (Even Function)`

35. Trigonometric Identities involving Cofunctions:

- `\( \sin\left(\frac{\pi}{2} - x\right) = \cos(x) \)`

- `\( \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \)`

36. General Solution of Trigonometric Equations:

- `\( \sin(x) = \sin(\alpha) \) has solutions \( x = \alpha + 2\pi n \) and \( x = \pi - \alpha + 2\pi n \)`

- `\( \cos(x) = \cos(\alpha) \) has solutions \( x = \alpha + 2\pi n \) and \( x = -\alpha + 2\pi n \)`

These additional trigonometric formulas and identities extend the range of problems that can be solved using trigonometry and provide deeper insights into the relationships between trigonometric functions.