Understanding the Given Expression
Given the expression:
sin(theta) = (a^2 - 1) / (a^2 + 1)
This equation relates the sine of an angle to a variable 'a'.
Finding the Value of Theta
To find the value of theta, we can use the inverse sine function (arcsin) of the expression:
- theta = arcsin((a^2 - 1) / (a^2 + 1))
This gives us theta in terms of 'a'.
Range of Sine Function
It's crucial to note that the sine function has a range of [-1, 1]. Therefore, we need to ensure that the expression falls within this range:
- The numerator (a^2 - 1) can take values from -1 to 1.
- The denominator (a^2 + 1) is always positive since a^2 is non-negative.
Analyzing the Expression
1. When a = 1:
- sin(theta) = (1^2 - 1) / (1^2 + 1) = 0, so theta = 0 degrees.
2. When a > 1:
- The numerator becomes positive, resulting in positive sine values.
3. When 0 < a="" />< />:
- The numerator becomes negative, resulting in negative sine values.
Conclusion
Thus, the value of theta can be expressed as:
- theta = arcsin((a^2 - 1) / (a^2 + 1)), provided that the value of sin(theta) remains between -1 and 1 based on the value of 'a'.
This relationship gives insights into how the angle theta varies with different values of 'a'. Always check the constraints of 'a' to ensure valid sine values.