Trigonometric Equations: Solution of trigonometric equations - based on extreme values of trigonometric ratios
Trigonometric Equations: Solution Based on Extreme Values of Trigonometric Ratios
Trigonometric equations are mathematical statements that involve trigonometric functions such as sine, cosine, tangent, etc. The solutions to these equations are the angles (usually measured in radians or degrees) that satisfy the equation. Understanding the extreme values of trigonometric ratios is crucial in solving these equations, as these values often correspond to the maximum or minimum points on the unit circle.
Extreme Values of Trigonometric Ratios
The trigonometric ratios, namely sine (sin), cosine (cos), and tangent (tan), have specific ranges and extreme values within the interval ([0, 2\pi]) or ([0^\circ, 360^\circ]).
| Trigonometric Function | Range | Extreme Values (Maximum) | Extreme Values (Minimum) |
|---|---|---|---|
| Sine (sin) | ([-1, 1]) | 1 at (\frac{\pi}{2}) or (90^\circ) | -1 at (\frac{3\pi}{2}) or (270^\circ) |
| Cosine (cos) | ([-1, 1]) | 1 at (0) or (0^\circ) | -1 at (\pi) or (180^\circ) |
| Tangent (tan) | ((-\infty, \infty)) | Undefined at (\frac{\pi}{2}) and (\frac{3\pi}{2}) or (90^\circ) and (270^\circ) | Undefined at (\frac{\pi}{2}) and (\frac{3\pi}{2}) or (90^\circ) and (270^\circ) |
Solution of Trigonometric Equations
When solving trigonometric equations, it is important to consider the periodic nature of the trigonometric functions. For example, the sine and cosine functions have a period of (2\pi) radians or (360^\circ), which means they repeat their values every (2\pi) radians or (360^\circ).
General Solutions
The general solution of a trigonometric equation takes into account all possible angles that satisfy the equation by adding integer multiples of the period. For sine and cosine, the general solutions can be written as:
For (\sin \theta = a), the general solution is: [ \theta = \begin{cases} \sin^{-1}(a) + 2k\pi, & \text{if } -1 \leq a \leq 1 \ \pi - \sin^{-1}(a) + 2k\pi, & \text{if } -1 \leq a \leq 1 \end{cases} ] where (k) is any integer.
For (\cos \theta = a), the general solution is: [ \theta = \begin{cases} \cos^{-1}(a) + 2k\pi, & \text{if } -1 \leq a \leq 1 \ -\cos^{-1}(a) + 2k\pi, & \text{if } -1 \leq a \leq 1 \end{cases} ] where (k) is any integer.
Examples
Let's look at some examples to illustrate how to solve trigonometric equations based on extreme values.
Example 1: Solve (\sin \theta = 1)
Since the extreme value of sine is 1 at (\theta = \frac{\pi}{2}), the solutions are:
[ \theta = \frac{\pi}{2} + 2k\pi ]
where (k) is any integer.
Example 2: Solve (\cos \theta = -1)
The extreme value of cosine is -1 at (\theta = \pi), so the solutions are:
[ \theta = \pi + 2k\pi ]
where (k) is any integer.
Example 3: Solve (\tan \theta = 0)
Tangent has a value of 0 at (\theta = 0) and (\theta = \pi). Since the period of tangent is (\pi), the general solution is:
[ \theta = k\pi ]
where (k) is any integer.
Conclusion
Solving trigonometric equations requires an understanding of the properties of trigonometric functions, including their periods and extreme values. By knowing these properties, one can determine the general solutions and find all possible angles that satisfy the given equation. Remember that the solutions can be expressed in radians or degrees, and it's important to consider the domain of the problem when finding the solutions.