Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.
Sagot :
Sure! Let's go through the steps to derive the cofunction identity \(\sin \left(\frac{\pi}{2} - x \right) = \cos(x)\).
### Step-by-Step Derivation
1. Starting Expression:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) \][/tex]
2. Use the angle addition formula for sine:
[tex]\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \][/tex]
Applying this formula where \(A = \frac{\pi}{2}\) and \(B = x\), we get:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = \sin \left( \frac{\pi}{2} \right) \cos(x) - \cos \left( \frac{\pi}{2} \right) \sin(x) \][/tex]
3. Evaluate \(\sin \left( \frac{\pi}{2} \right)\) and \(\cos \left( \frac{\pi}{2} \right)\):
From the unit circle or standard trigonometric values:
[tex]\[ \sin \left( \frac{\pi}{2} \right) = 1 \][/tex]
[tex]\[ \cos \left( \frac{\pi}{2} \right) = 0 \][/tex]
4. Substitute these values back into the expression:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = 1 \cdot \cos(x) - 0 \cdot \sin(x) \][/tex]
5. Simplify the expression:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = \cos(x) \][/tex]
Thus, we have derived the cofunction identity:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = \cos(x) \][/tex]
This completes the derivation of the cofunction identity.
### Step-by-Step Derivation
1. Starting Expression:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) \][/tex]
2. Use the angle addition formula for sine:
[tex]\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \][/tex]
Applying this formula where \(A = \frac{\pi}{2}\) and \(B = x\), we get:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = \sin \left( \frac{\pi}{2} \right) \cos(x) - \cos \left( \frac{\pi}{2} \right) \sin(x) \][/tex]
3. Evaluate \(\sin \left( \frac{\pi}{2} \right)\) and \(\cos \left( \frac{\pi}{2} \right)\):
From the unit circle or standard trigonometric values:
[tex]\[ \sin \left( \frac{\pi}{2} \right) = 1 \][/tex]
[tex]\[ \cos \left( \frac{\pi}{2} \right) = 0 \][/tex]
4. Substitute these values back into the expression:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = 1 \cdot \cos(x) - 0 \cdot \sin(x) \][/tex]
5. Simplify the expression:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = \cos(x) \][/tex]
Thus, we have derived the cofunction identity:
[tex]\[ \sin \left( \frac{\pi}{2} - x \right) = \cos(x) \][/tex]
This completes the derivation of the cofunction identity.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover insightful answers at IDNLearn.com. We appreciate your visit and look forward to assisting you again.