- Recall the sine angle subtraction formula: \(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\).
- Identify the trigonometric values \(\sin \left(\frac{\pi}{2}\right) = 1\) and \(\cos \left(\frac{\pi}{2}\right) = 0\).
- Substitute these values into the expression to get \((1)\cos(x) - (0)\sin(x)\).
- Simplify the expression to arrive at the final identity: \(\sin \left(\frac{\pi}{2}-x\right) = \boxed{\cos(x)}\).
Explanation
- Problem Analysis
The goal is to derive the trigonometric cofunction identity for \(\sin \left(\frac{\pi}{2}-x\right)\) using the sine angle subtraction formula. The formula states that for any angles \(A\) and \(B\), \(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\). We are given the identity \(\sin \left(\frac{\pi}{2}-x\right) = \sin \left(\frac{\pi}{2}\right) \cos (x) - \cos \left(\frac{\pi}{2}\right) \sin (x)\). We need to determine the values of \(\sin \left(\frac{\pi}{2}\right)\) and \(\cos \left(\frac{\pi}{2}\right)\) to complete the derivation.
- Evaluation and Substitution
Using the unit circle definition or standard trigonometric values, we know that at the angle \(\frac{\pi}{2}\) radians (or \(90^\circ\)):
- \(\sin \left(\frac{\pi}{2}\right) = 1\)
- \(\cos \left(\frac{\pi}{2}\right) = 0\)
These values are fundamental in trigonometry. Substituting these into our given expression:
\(\sin \left(\frac{\pi}{2}-x\right) = (1) \cos (x) - (0) \sin (x)\)
- Simplification and Conclusion
Now, we simplify the expression obtained by performing the multiplications:
- The first term is \(1 \times \cos(x) = \cos(x)\).
- The second term is \(0 \times \sin(x) = 0\).
Thus, the expression becomes \(\cos(x) - 0\), which simplifies to \(\cos(x)\).
Therefore, the cofunction identity is \(\sin \left(\frac{\pi}{2}-x\right) = \cos(x)\). The values to fill in the boxes are \(1\), \(0\), and \(\cos(x)\), respectively.
Examples
Understanding cofunction identities is crucial for simplifying complex trigonometric expressions and solving equations in engineering and physics. For instance, if a mechanical engineer is analyzing the phase shift in an alternating current circuit or the vibration modes of a structure, they often need to switch between sine and cosine representations to align different components of their model. Knowing that a sine wave can be converted into a cosine wave by a phase shift of \(\frac{\pi}{2}\) allows for the consolidation of terms, leading to more efficient calculations and cleaner designs.