To determine if the given trigonometric identity [tex]\(\cos \left(\frac{\pi}{2}-x\right)=\sin x\)[/tex] is true, let's analyze it step by step.
1. Understanding Co-Function Identities:
In trigonometry, there is a special relationship between the sine and cosine functions called the co-function identity. The co-function identities are derived from the nature of the unit circle and how the sine and cosine values are related based on their angles.
2. Specific Co-Function Identity:
One of the key co-function identities is:
[tex]\[
\cos\left(\frac{\pi}{2} - x\right) = \sin(x)
\][/tex]
This identity means that if you take the cosine of the complement of an angle [tex]\(x\)[/tex] (which is [tex]\(\frac{\pi}{2} - x\)[/tex]), it equals the sine of that angle.
3. Why is this True?
To understand why, consider the unit circle. The angle [tex]\(\frac{\pi}{2} - x\)[/tex] is complementary to [tex]\(x\)[/tex]. On the unit circle:
- The cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle.
- The sine of an angle is the y-coordinate of the same point.
When you subtract an angle from [tex]\(\frac{\pi}{2}\)[/tex] (which corresponds to 90 degrees), you switch the x and y coordinates because you're rotating by 90 degrees. Therefore:
- The x-coordinate (cosine) of the angle [tex]\(\frac{\pi}{2} - x\)[/tex] becomes the y-coordinate (sine) of the angle [tex]\(x\)[/tex].
4. Conclusion:
Given this understanding, we can see that the relationship [tex]\(\cos \left(\frac{\pi}{2}-x\right)=\sin x\)[/tex] is indeed true.
So the correct answer is:
A. True
