To verify the trigonometric identity [tex]\(\sin(180^\circ - \theta) = \sin(\theta)\)[/tex], we will follow a step-by-step approach and use some properties of trigonometric functions and their behavior.
### Step-by-Step Verification:
1. Understand the Identity in Question:
The identity we aim to verify is [tex]\(\sin(180^\circ - \theta) = \sin(\theta)\)[/tex].
2. Angle Difference Property of Sine:
Recall that sine is a function that depends on the angle provided. We need to recognize that [tex]\(\sin(180^\circ - \theta)\)[/tex] involves the sine of an angle subtraction.
3. Reference the Unit Circle:
Consider the unit circle definition of the sine function:
- On the unit circle, [tex]\(180^\circ\)[/tex] corresponds to the point [tex]\((-1, 0)\)[/tex].
- The angle [tex]\(\theta\)[/tex] can be plotted in standard position, rotating counterclockwise from the positive x-axis.
4. Property of Sine with Angle Subtraction:
The sine function has the property where [tex]\(\sin(180^\circ - \theta) = \sin(\theta)\)[/tex]. This is derived from the symmetrical properties of the sine function about the [tex]\( y \)[/tex]-axis.
5. Simplify the Left Hand Side:
Compute the left-hand side (LHS) of the given equation [tex]\(\sin(180^\circ - \theta)\)[/tex]:
- The angle subtraction formula ensures that subtracting an angle from [tex]\(180^\circ\)[/tex] maintains the original sine value.
6. Compare to the Right-Hand Side:
The right-hand side (RHS) of the equation is [tex]\(\sin(\theta)\)[/tex], which is straightforward and remains as it is.
### Final Simplification:
- Evaluate LHS:
[tex]\[
\sin(180^\circ - \theta) = \sin(\theta)
\][/tex]
- Evaluate RHS:
[tex]\[
\sin(\theta) = \sin(\theta)
\][/tex]
### Comparison and Conclusion:
After simplifying both sides, we obtain:
- LHS: [tex]\(\sin(\theta)\)[/tex]
- RHS: [tex]\(\sin(\theta)\)[/tex]
Since both sides are equal, the identity holds true.
Thus, we can conclude that [tex]\(\sin(180^\circ - \theta) = \sin(\theta)\)[/tex] is indeed a valid trigonometric identity.
