Resources | Subject Notes | Mathematics
We will explore some key trigonometric identities that are useful for simplifying expressions and solving equations.
| Formula | |
|---|---|
| $ \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)} $ |
| Formula | |
|---|---|
| $ \sin(\theta + \phi) = \sin(\theta) \cos(\phi) + \cos(\theta) \sin(\phi) $ | |
| $ \cos(\theta + \phi) = \cos(\theta) \cos(\phi) - \sin(\theta) \sin(\phi) $ | |
| $ \tan(\theta + \phi) = \frac{\tan(\theta) + \tan(\phi)}{1 - \tan(\theta) \tan(\phi)} $ | |
| $ \sin(\theta - \phi) = \sin(\theta) \cos(\phi) - \cos(\theta) \sin(\phi) $ | |
| $ \cos(\theta - \phi) = \cos(\theta) \cos(\phi) + \sin(\theta) \sin(\phi) $ | |
| $ \tan(\theta - \phi) = \frac{\tan(\theta) - \tan(\phi)}{1 + \tan(\theta) \tan(\phi)} $ |
To solve trigonometric equations, we use identities to simplify the equation into a form that can be solved. This often involves expressing the equation in terms of a single trigonometric function.
Example: Solve $ \cos(x) = 0 $ for $ 0 \le x < 360^\circ $.
Using the identity $ \cos(x) = 0 $ when $ x = 90^\circ $ and $ x = 270^\circ $.
Once we have found the general solutions to a trigonometric equation, we can find the solutions within a specific interval by adding or subtracting multiples of the period of the trigonometric function.
The period of $ \sin(x) $ and $ \cos(x) $ is $ 360^\circ $ or $ 2\pi $ radians.
Trigonometric identities are crucial for simplifying expressions and solving problems in various fields, including physics, engineering, and navigation.
For example, simplifying expressions involving square roots of trigonometric functions often requires using identities like $ \sin^2(\theta) + \cos^2(\theta) = 1 $.