To analyze Ingrid's claim and steps in verifying her claim, let's carefully review her process step by step.
First, recall the trigonometric identity for the cosine of a sum of angles:
[tex]\[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \][/tex]
Given Ingrid's claim is:
[tex]\[ \cos \left(-\frac{\pi}{2}+y\right) = \sin(y) \][/tex]
Let's break down her verification steps.
Step 1:
[tex]\[ \cos \left(-\frac{\pi}{2}+y\right) = \sin (y) \][/tex]
This is the initial claim we need to verify.
Step 2:
[tex]\[ \cos \left(-\frac{\pi}{2}+y\right) = \cos \left(-\frac{\pi}{2}\right) \cos(y) - \sin \left(-\frac{\pi}{2}\right) \sin(y) \][/tex]
Here, Ingrid applies the cosine sum identity correctly.
Step 3:
[tex]\[ \cos \left(-\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin \left(-\frac{\pi}{2}\right) = -1 \][/tex]
Let's substitute these values into the equation:
[tex]\[ 0 \cdot \cos(y) - (-1) \cdot \sin(y) \][/tex]
Step 4:
[tex]\[ 0 \cdot \cos(y) - 1 \cdot (-\sin(y)) = 0 - (-\sin(y)) = \sin(y) \][/tex]
This step simplifies the above expression correctly.
Step 5:
[tex]\[ \sin(y) = \sin(y) \][/tex]
This is a true statement, indicating that the initial claim is correct.
Thus, Ingrid's claim that [tex]\(\cos \left(-\frac{\pi}{2}+y\right) = \sin(y) \)[/tex] is correct, and her steps in verifying her claim are also correct.
Therefore, the correct statement is:
Ingrid's claim is correct, and her steps are correct.
